It follows that $\lambda_{P_{D}}=\frac{t-3}{t+1}$ and the set of
$P_{D}$-optimal vectors consists only of the uniform vector
$(\frac{1}{t},\ldots,\frac{1}{t})\in{\mathbb{S}}_{t}^{*}$.

Proof. Take any ${\bf x}\in{\mathbb{S}}_{t}^{*}$. In order to prove that
$f({\bf x})\leq\lambda_{1}$, we split the argument into two cases.

First, suppose that $\max\\{x_{i}:i\in[t]\\}\leq 1/2$. Here, we have by Lemma
29 that

$f({\bf
x})\leq\lambda_{\overline{D}}\left(\frac{1}{t},\dots,\frac{1}{t}\right)-\frac{2}{3}\sum_{i=1}^{t}\left(x_{i}-\frac{1}{t}\right)^{2}+\lambda_{1}\sum_{i=1}^{t}x_{i}^{3}=\lambda_{\overline{D}}+\sum_{i=1}^{t}g(x_{i}),$

where $g(x):=-\frac{2}{3}(x-\frac{1}{t})^{2}+\lambda_{1}x^{3}$ for
$x\in[0,\frac{1}{2}]$.

The second derivative of the cubic polynomial $g$ has zero at
$x_{0}:=\frac{2(t+1)}{9(t-3)}$ which is larger than $1/t$. (Recall that $t$
was assumed sufficiently large.) Thus function $g$ is concave on
$[0,\frac{1}{t}]$. Unfortunately, it is not concave on the whole interval
$[0,1/2]$ so we consider a different function $g^{*}$ which equals $g$ on
$[0,\frac{1}{t}]$ and whose graph on $[\frac{1}{t},\frac{1}{2}]$ is the line
tangent to the graph of $g$ at $1/t$, that is, we set

$g^{*}(x):=g\left(\frac{1}{t}\right)+g^{\prime}\left(\frac{1}{t}\right)\left(x-\frac{1}{t}\right),\quad\mbox{for
$x\in[1/t,1/2]$}.$

By above, $g^{*}$ is a concave function on $[0,1/2]$. Also, we have that
$g\leq g^{*}$ on this interval. Indeed, since the second derivative
$g^{\prime\prime}$ changes sign from negative to positive at $x_{0}>1/t$, it
is enough to check only that $g(1/2)\leq g^{*}(1/2)$ and routine calculations
give that

$g^{*}(1/2)-g(1/2)=\frac{(t-2)^{2}(t^{2}+t+36)}{24t^{3}(t+1)}>0,$

as desired. Thus, since $g^{*}$ is a concave majorant of $g$, we have that

$\frac{1}{t}\sum_{i=1}^{t}g(x_{i})\leq\frac{1}{t}\sum_{i=1}^{t}g^{*}(x_{i})\leq
g^{*}\left(\frac{x_{1}+\ldots+x_{t}}{t}\right)=g^{*}(1/t)=g(1/t).$

This gives that $f({\bf x})\leq\lambda_{\overline{D}}+tg(1/t)=\lambda_{1}$.
Moreover, if we have equality then $x_{1}=\ldots=x_{t}=1/t$ (because $g^{*}$
is strictly concave on $[0,1/t]$).

Thus it remains to consider the case when some $x_{i}$ is strictly larger than
$1/2$. Without loss of generality, assume that $x_{1}>1/2$. Here, we can bound

$f({\bf x})\leq
h(x_{1}):=3!\,x_{1}(1-x_{1})^{2}\frac{t-3}{2(t-1)}+3!\,\left(\frac{1-x_{1}}{t-1}\right)^{3}{t-1\choose
3}+\lambda_{1}\left(x_{1}^{3}+(1-x_{1})^{3}\right),$

where the stated three terms come from the following arguments. The first term
accounts for the triples containing $x_{1}$ in the Lagrange polynomial
$\lambda_{\overline{D}}({\bf x})$. The link graph $L_{D}(1)$ is just a perfect
matching $M$ on $\\{2,\dots,t\\}$ (because $D$ is a Steiner triple system) and
receives total weight $1-x_{i}$. As it is well-known (see e.g. [29]), the
Largrangian of a graph is maximised by putting the uniform weight on a maximum
clique which, for the complement $L_{\overline{D}}(1)$ of a perfect matching,
has size $s:=(t-1)/2$. Thus $(1-x_{1})^{-2}\sum_{ij\in
L_{D}(1)}x_{i}x_{j}\leq{s\choose 2}/s^{2}=\frac{s-1}{2s}=\frac{t-3}{2(t-1)}$,
giving the first term. The second term just upper bounds the Lagrangian of
${\overline{D}}-1={\overline{D}}[\\{2,\ldots,t\\}]$ by the Largrangian of the
complete $3$-graph on $t-1$ vertices, scaling the result by the cube of the
total weight $1-x_{1}$. The third term uses the fact that the sum of cubes of
non-negative entries with sum $1-x_{1}$ is maximised when we put all weight on
a single element.

The coefficient at $x^{3}$ in the cubic polynomial $h(x)$ is
$\frac{2t^{2}-10t+9}{(t-1)^{2}}>0$. Also, the derivative of $h$ has two roots,
which are $\pm 1/\sqrt{2}+o(1)$ as $t\to\infty$. Thus, for $t$ large, the
function $h$ first decreases and then increases on $[1/2,1]$. So, in order to
show that $f({\bf x})\leq\lambda_{1}$, it is enough to check the inequality
$h(x)\leq\lambda_{1}$ only at the points $x=1/2$ and $x=1$. There, the values
of $h(x)-\lambda_{1}$ are respectively
$-\frac{2t^{3}-20t^{2}+47t-27}{8(t-1)^{2}(t+1)}<0$ and $0$. Thus $f({\bf
x}_{1})\leq\lambda_{1}$ also in Case 2. Furthermore, the equality can only be
attained if $x_{1}=1$ (and $x_{2}=\ldots=x_{t}=0$); however, we exclude
standard basis vectors from ${\mathbb{S}}_{t}^{*}$. This proves the first part
of the lemma.

Using the proved inequality $f\leq\lambda_{1}$, one can show by a simple
induction on the number of levels that every $P$-construction on $n\to\infty$
vertices has edge density at most $\lambda_{1}+o(1)$. Thus
$\lambda_{P_{D}}=\lambda_{1}$. Also, the set of $P_{D}$-extremal vectors,
which by definition consists of ${\bf x}\in{\mathbb{S}}_{t}^{*}$ with $f({\bf
x})=\lambda_{1}$, is exactly as claimed.

In the rest of the this section, whenever we have any
$I\subseteq\mathcal{ST\\!S}_{t}$, a set of Steiner triple systems on $[t]$, we
denote $P_{I}:=\\{P_{D}:D\in I\\}$. (Recall that
$P_{D}=(t,\overline{D},[t])$.) Also, let us call a partition
$V=V_{1}\cup\ldots\cup V_{t}$ balanced if for all $i,j\in[t]$ we have
$\left|\,|V_{i}|-|V_{j}|\,\right|\leq 1$.

###### Lemma 31

Let $t$ be sufficiently large and take any non-empty
$I\subseteq\mathcal{ST\\!S}_{t}$. Then there is $n_{0}=n_{0}(I)$ such that
every maximum $P_{I}$-mixing construction $G$ on $n\geq n_{0}$ vertices has
its bottom partition balanced.

Proof. View $n$ as tending to $\infty$ and $G$ be any maximum $P_{I}$-mixing
construction on $[n]$. Let $G$ have the base $P_{D}$ and the bottom partition
$[n]=V_{1}\cup\ldots\cup V_{t}$. Let $v_{i}:=|V_{i}|$ and $e_{i}:=|G[V_{i}]|$
for $i\in[t]$. By Lemma 30, we have $v_{i}=(1/t+o(1))n$ for each $i\in[t]$.

Suppose on the contrary that some two part sizes differ by more than 1, say,
$v_{1}\geq v_{2}+2$. Let $u$ (resp. $w$) be a vertex of minimum degree in
$G[V_{1}]$ (resp. maximum degree in $G[V_{2}]$). Let $G^{\prime}$ be obtained
from $G$ by removing $u$ and adding a double $w^{\prime}$ of $w$. Thus
$G^{\prime}$ is also a $P_{I}$-mixing construction, with the bottom parts
$V_{1}^{\prime}:=V_{1}\setminus\\{u\\}$,
$V_{2}^{\prime}:=V_{2}\cup\\{w^{\prime}\\}$, and $V_{i}^{\prime}:=V_{i}$ for
$3\leq i\leq t$. By the maximality of $G$, we have $|G|\geq|G^{\prime}|.$

Let us estimate $|G^{\prime}|-|G|$. The contribution from the edges that
intersect both the first part and the second part is

$\left((v_{1}-1)(v_{2}+1)-v_{1}v_{2}\right)\sum_{i\colon\\{1,2,i\\}\in{\overline{D}}}|V_{i}|=(v_{1}-v_{2}-1)\left(\frac{t-3}{t}+o(1)\right)n,$

where the equality uses Lemma 30 and the fact that there are exactly $t-3$
triples containing the pair $\\{1,2\\}$. By the maximality of $G$, each part
$V_{i}$ spans a maximum $P_{I}$-mixing construction; thus
$e_{i}=\Lambda_{P_{I}}(v_{i})$. Since $v_{1}\geq v_{2}$, we have by Lemma 13
that $\Lambda_{P_{I}}(v_{1})/{v_{1}\choose
3}\leq\Lambda_{P_{I}}(v_{2})/{v_{2}\choose 3}$. Thus the degree of $u$ in
$G[V_{1}]$ is at most

$\frac{3e_{1}}{v_{1}}=\frac{3\Lambda_{P_{I}}(v_{1})}{v_{1}}\leq\frac{3\Lambda_{P_{I}}(v_{2})\,{v_{1}\choose
3}}{v_{1}{v_{2}\choose
3}}=\frac{3e_{2}}{v_{2}}\cdot\frac{(v_{1}-1)(v_{1}-2)}{(v_{2}-1)(v_{2}-2)}.$

The degree of $w^{\prime}$ in $G^{\prime}$ is at least the degree of $w$ in
$G$ which in turn is at least $3e_{2}/v_{2}$. Thus the contribution to
$|G^{\prime}|-|G|$ of edges inside the first or second part is at least

$\frac{3e_{2}}{v_{2}}-\frac{3e_{1}}{v_{1}}\geq\frac{3e_{2}}{v_{2}}\left(1-\frac{(v_{1}-1)(v_{1}-2)}{(v_{2}-1)(v_{2}-2)}\right)=-\frac{3e_{2}}{v_{2}}\cdot\frac{(v_{1}-v_{2})(v_{1}+v_{2}-3)}{(v_{2}-1)(v_{2}-2)}.$

This is, by $e_{2}=(\lambda_{P_{D}}+o(1)){v_{2}\choose
3}=\left(\frac{t-3}{t+1}+o(1)\right){n/t\choose 3}$,

$(-1+o(1))\,\frac{3\,\frac{t-3}{t+1}\,{n/t\choose
3}}{n/t}\cdot\frac{(v_{1}-v_{2})\,2n/t}{(n/t)^{2}}=-\frac{(v_{1}-v_{2})(t-3)n}{t(t+1)}+o((v_{1}-v_{2})n).$

By putting both contributions together and using
$v_{1}-v_{2}-1\geq(v_{1}-v_{2})/2$, we obtain that

$\displaystyle 0$ $\displaystyle\geq$ $\displaystyle|G^{\prime}|-|G|\ \geq\
(v_{1}-v_{2}-1)\,\frac{t-3}{t}n-\frac{(v_{1}-v_{2})(t-3)n}{t(t+1)}+o((v_{1}-v_{2})n)$
$\displaystyle\geq$
$\displaystyle(v_{1}-v_{2})\,\frac{(t-3)n}{t}\left(\frac{1}{2}-\frac{1}{t+1}\right)+o((v_{1}-v_{2})n)\
>\ 0,$

which is the desired contradiction.

For every $D\in\mathcal{ST\\!S}_{t}$, we construct a parameter $F(D)$ of much
lower complexity than $D$ that nonetheless contains enough information to
compute the sizes of maximum balanced $\overline{D}$-blowups of all large
orders. More precisely, we do the following for every
$q\in\\{0,\ldots,t-1\\}$. Let $\ell\to\infty$. For every $q$-set
$Q\subseteq[t]$, consider a $\overline{D}$-blowup $G$ on $[t\ell+q]$ with
partition $V_{1}\cup\ldots\cup V_{t}$, where

$|V_{i}|=\left\\{\begin{array}[]{ll}\ell+1,&\mbox{if $i\in Q$,}\\\
\ell,&\mbox{if $i\in[t]\setminus Q$}.\end{array}\right.$

Thus, the $q$ larger parts are exactly those specified by $Q$. Clearly, the
size of $G$ is

$p_{D,Q}(\ell):=(\ell+1)^{3}t_{D,Q,3}+\ell(\ell+1)^{2}t_{D,Q,2}+\ell^{2}(\ell+1)t_{D,Q,1}+\ell^{3}\left({t\choose
3}-t_{D,Q,3}-t_{D,Q,2}-t_{D,Q,1}\right),$

where, for $i\in[3]$, we let $t_{D,Q,i}$ denote the number of triples in
$\overline{D}$ that have exactly $i$ vertices in $Q$. This is a polynomial
function of $\ell$. If we take another $q$-set $Q^{\prime}$ then the
polynomial $p_{D,Q}(\ell)-p_{D,Q^{\prime}}(\ell)$ does not change sign for
large $\ell$ (possibly being the zero polynomial). Since there are finitely
many choices of $Q$ (namely, ${t\choose q}$) there are $Q_{D,q}\in{[t]\choose
q}$ and $\ell_{0}$ such that

$p_{D,Q_{D,q}}(\ell)\geq p_{D,Q^{\prime}}(\ell),\quad\mbox{for all
$Q^{\prime}\in{[t]\choose q}$ and $\ell\geq\ell_{0}$}.$ (36)

Fix one such $Q_{D,q}$ for each $q\in\\{0,\dots,t-1\\}$ and define

$F(D):=((t_{D,Q_{D,q},i})_{i=1}^{3})_{q=0}^{t-1}.$ (37)

The number of possible values of $F$ is upper bounded by, say, $t^{9t}$
because each individual $t_{D,Q,i}$ assumes at most ${t\choose 3}+1\leq t^{3}$
possible values.

###### Lemma 32

Let $t$ be sufficiently large. Let $I\subseteq\mathcal{ST\\!S}_{t}$ be any
subset on which the above function $F$ is constant. Then there is $n_{0}$ such
that for all $n\geq n_{0}$ and all $D\in I$ there is a maximum $P_{I}$-mixing
construction $G$ on $[n]$ with the base $P_{D}$.

Proof. Choose $n_{0}$ sufficiently large, in particular so that, for every
$n\geq n_{0}$, (36) holds for every $D\in I$ with respect to $\ell:=\lfloor
n/t\rfloor$ and $q:=n-t\ell$.

Take any $n\geq n_{0}$ and $D\in I$. We have to exhibit a maximum
$P_{I}$-mixing construction $G$ with base $P_{D}$. Let $\ell:=\lfloor
n/t\rfloor$, $q:=n-t\ell$, and $Q:=Q_{D,q}$. Take a balanced partition
$V_{1}\cup\ldots\cup V_{t}$ of $[n]$ with $|V_{j}|=\ell+1$ exactly for $j\in
Q$. Let $G$ be obtained from ${\overline{D}}(\\!(V_{1},\ldots,V_{t})\\!)$ by
adding for every $j\in[t]$ a maximum $P_{I}$-mixing construction on $V_{j}$.

Let us show that the $P_{I}$-mixing construction $G$ is maximum. Note that the
size of the graph $G$ is

$|G|=p_{D,Q}(\ell)+q\,\Lambda_{P_{I}}(\ell+1)+(t-q)\,\Lambda_{P_{I}}(\ell).$
(38)

Let $G^{\prime}$ be any maximum $P_{I}$-mixing construction on $[n]$. Let
$G^{\prime}$ have the base $D^{\prime}$ and the bottom partition
$V_{1}^{\prime}\cup\dots\cup V_{t}^{\prime}$. This partition must be balanced
by Lemma 31, since $n$ is large. Let $Q^{\prime}\in{[t]\choose q}$ consist of
the indices of parts of size $\ell+1$. Clearly, $|Q^{\prime}|=q$. By the
maximality of $G^{\prime}$, every part $V_{s}^{\prime}$, $s\in[t]$, induces a
maximum $P_{I}$-mixing construction. Thus the obvious analogue of (38) holds
for $G^{\prime}$ as well. Let $Q^{\prime\prime}:=Q_{D^{\prime},q}$. Since
$\ell\geq\lfloor n_{0}/t\rfloor$ is large, we have
$p_{D^{\prime},Q^{\prime}}(\ell)\leq p_{D^{\prime},Q^{\prime\prime}}(\ell)$.
Since $F(D)=F(D^{\prime})$, the polynomials $p_{D^{\prime},Q^{\prime\prime}}$
and $p_{D,Q}$ are the same. Putting all together, we obtain

$|G^{\prime}|-|G|=p_{D^{\prime},Q^{\prime}}(\ell)-p_{D,Q}(\ell)\leq
p_{D^{\prime},Q^{\prime\prime}}(\ell)-p_{D,Q}(\ell)=0.$

Thus $|G|$ is indeed a maximum $P_{I}$-mixing construction.

Let us remark that Lemma 32 need not hold for small $n$ when it is in general
possible that some of the patterns $P_{D}$ for $D\in I$ cannot be the base in
a maximum $P_{I}$-mixing construction.

Proof of Theorem 3. Keevash [20, Theorem 2.2] proved that if $t\to\infty$ is 1
or 3 modulo 6 then the number of Steiner triples systems on $[t]$ is
$(t/\mathrm{e}^{2}+o(1))^{t^{2}/6}$. Note that the function $F$ assumes at
most $t^{9t}$ values while each isomorphism class of $\mathcal{ST\\!S}_{t}$
has at most $t!$ elements. Thus we can fix a sufficiently large $t$ and a
subset $I\subseteq\mathcal{ST\\!S}_{t}$ consisting of non-isomorphic 3-graphs
such that $F$ is constant on $I$ while

$|I|\geq\frac{(t/\mathrm{e}^{2}+o(1))^{t^{2}/6}}{t!\,t^{9t}}>t!.$

Let ${\cal F}$ be the finite family of $3$-graphs returned by Theorem 2; thus
maximum ${\cal F}$-free $3$-graphs are exactly maximum $P_{I}$-mixing
constructions. Let us show that this family ${\cal F}$ satisfies both parts of
Theorem 3. Let $n_{0}$ be sufficiently large.

Given any $n\in\mathbb{N}$, let $V:=[n]$ and consider the family ${\cal P}$ of
3-graphs $G$ that can be recursively constructed as follows. If the current
vertex set $V$ has less than $n_{0}$ vertices, put any maximum $P_{I}$-mixing
construction on $V$ and stop. So suppose that $|V|\geq n_{0}$. Pick any $D\in
I$. Let $G^{\prime}$ be a maximum $P_{I}$-mixing construction on $V$ with the
base $P_{D}$, which exists by Lemma 32. Let $V=V_{1}\cup\ldots\cup V_{t}$ be
the bottom partition of $G^{\prime}$. Let $G$ be obtained by taking all bottom
edges of $G^{\prime}$ and adding for each $i\in[t]$ a $3$-graph on $V_{i}$
that can be recursively constructed by the above procedure.

Let us observe some easy properties of any obtained $G\in{\cal P}$. By
definition, $G$ is a $P_{I}$-mixing construction. In fact, it is a maximum
one, which can be shown by induction on the number of vertices: the initial
$P_{I}$-mixing construction $G^{\prime}$ is maximum and when we “erase” edges
in $G^{\prime}[V_{i}]$, we add back the same number of edges by induction.

Let the size-$m$ truncated tree ${\bf T}_{G}^{\mathrm{size}\geq m}$ of
$P_{I}$-mixing construction $G$ be the subtree of ${\bf T}_{G}$ where we keep
only those nodes that corresponds to parts in ${\bf V}_{G}$ of size at least
$m$. Of course, if a node is not included into ${\bf T}_{G}^{\mathrm{size}\geq
m}$ then all its descendants are not, so ${\bf T}_{G}^{\mathrm{size}\geq m}$
is a subtree of ${\bf T}_{G}$.

In the rest of Section 4, we would need to work with unordered trees where the
$t$ children of a non-leaf vertex are not ordered and the first part ${\bf i}$
of each note $({\bf i},x)$, that can be used to order children, is erased (but
we keep the second part $x$). For these objects we will use the (non-bold)
symbol $T$ instead of ${\bf T}$. In particular, the size-$n_{0}$ truncated
tree $T_{G}^{\mathrm{size}\geq n_{0}}$ is the unordered subtree of the tree of
the $P_{I}$-mixing construction $G$, where we keep only those nodes that
corresponds to parts of size at least $n_{0}$. Suppose now that $G\in\Sigma
P_{I}$ is maximum. Then every branch of $T_{G}^{\mathrm{size}\geq n_{0}}$ has
length at least, say, $\log_{t+1}(n/n_{0})$, because when we subdivide any set
$V$ with at least $n_{0}$ vertices, its partition is balanced by Lemma 31 and
thus the smallest part size is at least $\lfloor|V|/t\rfloor\geq|V|/(t+1)$.
Thus any resulting tree has at least $s:=t^{\log_{t+1}(n/n_{0})-1}$ non-leaf
vertices. Since the children of every non-leaf can be labelled by any element
of $I$ (as long as we use the same element for all children), we have at least
$|I|^{s}$ choices here. The number of ways that an isomorphic copy of any
feasible $I$-labelled rooted tree $T$ can be generated as above, rather
roughly, is most $(t!)^{s}$. Thus there are at least $|(I|/t!)^{s}$ non-
isomorphic (unordered $I$-labelled rooted) trees obtainable this way. This is
exponential in $n$ (since $n_{0}$ is fixed); thus the first part of Theorem 3
follows from the following claim.

###### Claim 32.1

If $G,G^{\prime}\in{\cal P}$ are isomorphic $3$-graphs then their
$n_{0}$-truncated unordered trees $T_{G}^{\mathrm{size}\geq n_{0}}$ and
$T_{G}^{\mathrm{size}\geq n_{0}}$ are isomorphic.

Proof of Claim. We use induction on $n$, the number of vertices in $G$. If
$n<n_{0}$, then the truncated trees of $G$ and $G^{\prime}$ are empty and thus
the conclusion vacuously holds. Suppose $n\geq n_{0}$ and that the identity
map on $[n]$ gives an isomorphism between some $G,G^{\prime}\in{\cal P}$.

By Lemmas 17 and 22, the maximum $P_{I}$-mixing construction $G$ is rigid. (In
fact, the proof of Lemma 22 simplifies greatly in this case and every
$P_{I}$-mixing construction $G$ with all bottom parts non-empty is rigid since
we can identify bottom edges of $G$ as precisely those that do not contain the
symmetric difference of some two distinct edges of $G$.)

Thus, $G$ and $G^{\prime}$ have the same base $P_{D}$ and the isomorphism
between their bottom edge-sets,
$\overline{D}(\\!(V_{1},\ldots,V_{t})\\!)\subseteq G$ and
$\overline{D}(\\!(V_{1}^{\prime},\ldots,V_{t}^{\prime})\\!)\subseteq
G^{\prime}$, comes from an automorphism $h$ of $D$, that is,
$V_{h(i)}^{\prime}=V_{i}$ for every $i\in[t]$. Now, using the automorphism $h$
relabel the bottom parts of $G^{\prime}$ so that $V_{i}^{\prime}=V_{i}$ for
$i\in[t]$ and apply induction to each pair $G[V_{i}],G^{\prime}[V_{i}]\in{\cal
P}$ of isomorphic (in fact, equal) $3$-graphs.

We now turn to the second part of the theorem. Take any $s$. Let $\ell$
satisfy $(|I|/t!)^{\ell-1}\geq s$. Fix sufficiently small $\epsilon>0$ and let
$n\to\infty$. It is enough to find $s$ maximum $P_{I}$-mixing constructions on
$[n]$ so that every two are at edit distance at least $\epsilon n^{3}$.

Call two $I$-labelled unordered trees $T$ and $T^{\prime}$ isomorphic up to
level $m$ if their first $m$ levels span isomorphic trees. For example, two
trees are isomorphic up to level 1 if and only the children of the root are
labelled by the same element of $I$ in the both trees. (Recall that the root
always get label $\emptyset$ while all children of a node always get the same
label.) For convenience, let us agree that every two trees are isomorphic up
to level 0.

By our choice of $\ell$, there are $s$ trees that are pairwise non-isomorphic
up to level $\ell$. Since $n$ is sufficiently large, we can assume by Lemma 32
that for each of these trees $T$ there is a maximum $P_{I}$-mixing
construction $G$ on $[n]$ whose unordered tree is isomorphic to $T$ up to
level $\ell$, that is, $T_{G}^{\mathrm{level}\leq\ell}\cong T$. Thus it is
enough to show that any two of the obtained $3$-graphs, say $G$ and
$G^{\prime}$ with unordered trees $T$ and $T^{\prime}$ respectively where
$T^{\prime}$ and $T^{\prime}$ are non-isomorphic up to level $\ell$, are at
the edit distance at least $\epsilon n^{3}$. Suppose that this is false and
the identity map exhibits this, that is, $|G\bigtriangleup
G^{\prime}|<\epsilon n^{3}$. Let ${\bf V}$ and ${\bf V}^{\prime}$ be the
partition structures of $G$ and $G^{\prime}$ respectively. Also, for a tree
$T$ and a sequence $(i_{1},\dots,i_{m})$, let $T[i_{1},\dots,i_{m}]$ be the
subtree formed by the vertex with the first coordinate $(i_{1},\dots,i_{m})$
and all its descendants in $T$.

###### Claim 32.2

For every $m\leq\ell$, there are sequences
$(i_{1},\ldots,i_{m}),(i_{1}^{\prime},\ldots,i_{m}^{\prime})\in[t]^{m}$ such
that $T[i_{1},\dots,i_{m}]$ and
$T^{\prime}[i_{1}^{\prime},\ldots,i_{m}^{\prime}]$ are non-isomorphic up to
level $\ell-m$ and

$\left|V_{i_{1},\ldots,i_{m}}\bigtriangleup
V^{\prime}_{i_{1}^{\prime},\ldots,i_{m}^{\prime}}\right|\leq
2(t-1)m\cdot\epsilon n.$

Proof of Claim. We use induction on $m$ with the base case $m=0$ satisfied by
the empty sequences. Let $m\geq 1$ and suppose that we already constructed
some sequences
$(i_{1},\ldots,i_{m-1}),(i_{1}^{\prime},\ldots,i_{m-1}^{\prime})\in[t]^{m-1}$
that satisfy the claim for $m-1$. Let $U:=V_{i_{1},\ldots,i_{m-1}}$ and
$U^{\prime}:=V^{\prime}_{i_{1}^{\prime},\ldots,i_{m-1}^{\prime}}$. For
$j\in[t]$, let $U_{j}:=V_{i_{1},\ldots,i_{m-1},j}$ and
$U^{\prime}_{j}:=V^{\prime}_{i_{1}^{\prime},\ldots,i_{m-1}^{\prime},j}$. In
other words, $U=U_{1}\cup\dots\cup U_{t}$ and
$U^{\prime}:=U^{\prime}_{1}\cup\dots\cup U^{\prime}_{t}$ are the bottom
partitions of the $P_{I}$-mixing constructions $G[U]$ and
$G^{\prime}[U^{\prime}]$ respectively. These constructions are maximum; thus,
by Lemma 31, their bottom partitions are balanced and, by simple induction on
$m$, each part has size $n/t^{m}+O(1)$.

For each $i\in[t]$, there cannot be distinct $j,k\in[t]$ with each of
$A:=U_{i}\cap U^{\prime}_{j}$ and $B:=U_{i}\cap U^{\prime}_{k}$ having at
least $\epsilon n$ elements. Indeed, otherwise the co-degree of each pair
$(a,b)\in A\times B$ is at most $|U_{i}|-2=n/t^{m}+O(1)$ in $G[U]$ as all such
edges lie inside $U_{i}$, and at least $(t-3)n/t^{m}+O(1)$ in
$G^{\prime}[U^{\prime}]$ (since the bottom pattern of $G^{\prime}[U^{\prime}]$
has $t-3$ triples containing the pair $\\{i,j\\}$). This is impossible, since
then

$|G\bigtriangleup G^{\prime}|\geq|G^{\prime}\setminus
G|\geq|A|\cdot|B|\cdot\left(\frac{(t-4)n}{t^{m}}+O(1)\right)\geq\epsilon
n^{3}.$ (39)

Thus there is $h:[t]\to[t]$ with $|U_{i}\cap U^{\prime}_{j}|<\epsilon n$ for
each $j\in[t]$ different from $h(i)$. Since $\varepsilon\ll 1/t^{\ell}\leq
1/t^{m}$, we have

$|U_{i}\cap U^{\prime}_{h(i)}|\geq|U_{i}|-(t-1)\epsilon n-|U^{\prime}\setminus
U|\geq\frac{n}{t^{m}}-2(t-1)m\cdot\epsilon n+O(1),$ (40)

which is strictly larger than half of $|U^{\prime}_{h(i)}|$. Thus $h$ is an
injection and, by the equal finite cardinality of its domain and its range, a
bijection.

The map $h$ has to be an isomorphism between the base patters $D$ and
$D^{\prime}$ of $G[U]$ and $G^{\prime}[U^{\prime}]$ respectively. Indeed, if
$h$ does not preserve the adjacency for at least one triple then (40) implies
that, say, $|G\bigtriangleup G^{\prime}|\geq(n/(2t^{m}))^{3}>\epsilon n^{3}$,
a contradiction.

Since the trees $T[i_{1},\dots,i_{m-1}]$ and
$T^{\prime}[i_{1}^{\prime},\ldots,i_{m-1}^{\prime}]$ (which are the trees of
$G[U]$ and $G^{\prime}[U^{\prime}]$) are non-isomorphic up to level
$\ell-m+1$, there must be $i$ such that, letting $i_{m}:=i$ and
$i_{m}^{\prime}:=h(i)$, the trees $T[i_{1},\dots,i_{m}]$ and
$T^{\prime}[i_{1}^{\prime},\ldots,i_{m}^{\prime}]$ are non-isomorphic up to
level $\ell-m$. Also, by (40) (and its version when the roles of $G$ and
$G^{\prime}$ are exchanged), we have that

$\left|V_{i_{1},\dots,i_{m}}\bigtriangleup
V^{\prime}_{i_{1}^{\prime},\ldots,i_{m}^{\prime}}\right|\leq|U\bigtriangleup
U^{\prime}|+2(t-1)\cdot\epsilon n\leq 2(t-1)m\cdot\epsilon n.$

Thus the obtained sequences
$(i_{1},\ldots,i_{m}),(i_{1}^{\prime},\ldots,i_{m}^{\prime})\in[t]^{m}$
satisfy the claim.

Finally, a desired contradiction follows by taking $m:=\ell$ in the above
claim (as every two trees are isomorphic up to level 0), finishing the proof
of the second part of Theorem 3.

##  5 Feasible region

In this section we prove Theorem 5. We need some auxiliary results first.

###### Lemma 33

Suppose that $P=(m,E,R)$ is a minimal pattern. Then every pair
$\\{i,j\\}\in\binom{[m]}{2}$ is contained in some multiset in $E$.

Proof. Suppose to the contrary that there exists a pair
$\\{i,j\\}\in\binom{[m]}{2}$ that is not contained in any multiset in $E$. Let
$\lambda:=\lambda_{P}$. For ${\bf x}\in{\mathbb{S}}_{m}$, let $f({\bf
x}):=\lambda_{E}({\bf x})+\lambda\sum_{t\in R}x_{t}^{r}$. Note that no term in
the polynomial $f$ contains the product $x_{i}x_{j}$. Let ${\bf x}\in{\cal X}$
be an optimal vector for $P$. Let
$d_{i}:=\lambda_{L_{E}(i)}(x_{1},\ldots,x_{i-1},x_{i+1},\ldots,x_{m})$ and
$d_{j}:=\lambda_{L_{E}(j)}(x_{1},\ldots,x_{j-1},x_{j+1},\ldots,x_{m})$.

Suppose first that $i,j$ are both contained in $R$. Without loss of generality
we may assume that $d_{i}+\lambda x_{i}^{r-1}\geq d_{j}+\lambda x_{j}^{r-1}$.
Define the new vector

${\bf
y}:=(x_{1},\ldots,x_{i}+x_{j},\ldots,x_{j-1},0,x_{j+1},\ldots,x_{m})\in\mathbb{S}_{m}.$
(41)

It satisfies

$\displaystyle f({\bf y})-f({\bf x})$ $\displaystyle=$
$\displaystyle(x_{i}+x_{j})(d_{i}+\lambda(x_{i}+x_{j})^{r-1})-\left(x_{i}(d_{i}+\lambda
x_{i}^{r-1})+x_{j}(d_{j}+\lambda x_{j}^{r-1})\right)$ $\displaystyle\geq$
$\displaystyle(x_{i}+x_{j})(d_{i}+\lambda(x_{i}+x_{j})^{r-1})-(x_{i}+x_{j})(d_{i}+\lambda
x_{i}^{r-1}))$ $\displaystyle=$
$\displaystyle(x_{i}+x_{j})\left(\lambda(x_{i}+x_{j})^{r-1}-\lambda
x_{i}^{r-1}\right)>0,$

which contradicts our choice that ${\bf x}\in{\cal X}$.

Suppose that $i,j$ are both contained in $[m]\setminus R$. Without loss of
generality we may assume that $d_{i}\geq d_{j}$. Then, with ${\bf y}$ as in
(41), we have

$\displaystyle f({\bf y})-f({\bf x})$ $\displaystyle=$
$\displaystyle(x_{i}+x_{j})d_{i}-\left(x_{i}d_{i}+x_{j}d_{j}\right)\ =\
x_{j}(d_{i}-d_{j})\ \geq\ 0.$

This implies that ${\bf y}\in{\cal X}$, which contradicts Lemma 9(c).

Now, by symmetry, we may assume that $i\in R$ and $j\in[m]\setminus R$. Since
${\bf x}\in{\cal X}$, it follows from Lemma 9(d) that $\frac{\partial
f}{\partial_{i}}({\bf x})=r\cdot\lambda=\frac{\partial f}{\partial_{j}}({\bf
x})$, which implies that $d_{i}+r\lambda x_{i}^{r-1}=d_{j}$. Then, with ${\bf
y}$ as in (41), we have

$\displaystyle f({\bf y})-f({\bf x})$ $\displaystyle=$
$\displaystyle\left(\lambda_{E}({\bf y})+\lambda\sum_{t\in
R}y_{t}^{r}\right)-\left(\lambda_{E}({\bf x})+\lambda\sum_{t\in
R}x_{t}^{r}\right)$ $\displaystyle=$
$\displaystyle(x_{i}+x_{j})(d_{i}+\lambda(x_{i}+x_{j})^{r-1})-\left(x_{i}(d_{i}+\lambda
x_{i}^{r-1})+x_{j}d_{j}\right)$ $\displaystyle=$
$\displaystyle(x_{i}+x_{j})(d_{i}+\lambda(x_{i}+x_{j})^{r-1})-(x_{i}+x_{j})(d_{i}+\lambda
x_{i}^{r-1})$ $\displaystyle=$
$\displaystyle(x_{i}+x_{j})(\lambda(x_{i}+x_{j})^{r-1}-\lambda x_{i}^{r-1})\
>\ 0,$

which contradicts our choice that ${\bf x}\in{\cal X}$.

For every $i\in I$ and pattern $P_{i}=(m_{i},E_{i},R_{i})$ we let
$S_{i}\subseteq[m_{i}]\setminus R_{i}$ be the collection of
$j\in[m_{i}]\setminus R_{i}$ such that for every blowup
$E_{i}(\\!(V_{1},\ldots,V_{m_{i}})\\!)$ of $E_{i}$ the shadow of
$E_{i}(\\!(V_{1},\ldots,V_{m_{i}})\\!)$ has no edge inside $V_{j}$.

###### Lemma 34

For every finite collection $P_{I}$ of minimal 3-uniform patterns with the
same Lagrangian $\lambda\in(0,1)$, there is an integer $M$ such that for every
$\epsilon>0$ there exist $\delta>0$ and $N_{0}>0$ such that the following
holds for all $n\geq N_{0}$. If $G$ is an $\mathcal{F}_{M}$-free $3$-graph on
$[n]$ with at least $(\lambda-\delta)\binom{n}{3}$ edges, then there exists a
$P_{I}$-mixing construction $H$ on $[n]$ with base $P_{i}$ and bottom
partition $V(G)=V_{1}\cup\cdots\cup V_{m_{i}}$ for some $i\in I$ such that
$\delta(H)\geq(\lambda-\epsilon)\binom{n-1}{2}$, $|H\bigtriangleup
G|\leq\epsilon{n\choose 3}$, $(|V_{1}|/n,\dots,|V_{m_{i}}|/n)$ is
$\epsilon$-close to a $P_{i}$-optimal vector, and $\sum_{j\in S_{i}}|(\partial
G)[V_{j}]|\leq\epsilon{n\choose 2}$.

Proof. Many steps of this proof are similar to the analogous parts of the
proof of Lemma 25. So we may be brief when the appropriate adaptation is
straightforward,

Let $\ell_{0}$ be the constant returned by Lemma 24 and then let $M$ be
sufficiently large. Given $\epsilon>0$, choose sufficiently small positive
constants in this order:
$\delta_{4}\gg\delta_{3}\gg\delta_{2}\gg\delta_{1}\gg\delta$. Let $n\to\infty$
and let $G$ be any ${\cal F}_{M}$-free $r$-graph on $V:=[n]$ with at least
$(\lambda-\delta){n\choose 3}$ edges.

Due to Theorem 2(b), we may assume that $|G\bigtriangleup
H|\leq\delta_{1}{n\choose 3}$ for some $P_{I}$-mixing construction $H$. Let
$H$ have the partition structure ${\bf V}$, the base $P_{i}$ and the bottom
partition $V_{1},\ldots,V_{m_{i}}$. By modifying $H$ as in the argument
leading to (28), we can further assume that

$\delta(H)\geq(\lambda-\delta_{2})\binom{n-1}{2}\quad\mbox{and}\quad|G\bigtriangleup
H|\leq\delta_{2}{n\choose 3}.$ (42)

By Lemma 19(b), the bottom part ratios are within $\epsilon$ from a
$P_{i}$-optimal vector. Thus this $H$ satisfies the first three properties
stated in the lemma. The rest of the proof is dedicated to proving the
remaining property that the total shadow of $G$ inside the parts $V_{j}$ with
$j\in S_{i}$ is “small”.

Let

$\displaystyle Z_{1}:=\left\\{v\in V\colon
d_{G}(v)\leq(\lambda-\delta_{3})\binom{n-1}{2}\right\\}\quad\text{and}\quad
G_{1}:=G-Z_{1}.$

By (42), we have the following inequalities.

###### Claim 34.1

It holds that

$\displaystyle|Z_{1}|$ $\displaystyle\leq$
$\displaystyle\frac{3|G\bigtriangleup H|}{(\delta_{3}-\delta_{2}){n-1\choose
2}}\ \leq\ \delta_{3}n,$ $\displaystyle|G_{1}|$ $\displaystyle\geq$
$\displaystyle|G|-|Z_{1}|{n-1\choose 2}\geq(\lambda-\delta){n\choose
3}-\delta_{3}n{n-1\choose 2}\ \geq\ \lambda\binom{n}{3}-\delta_{3}n^{3},$
$\displaystyle\delta(G_{1})$ $\displaystyle\geq$
$\displaystyle(\lambda-\delta_{3})\binom{n-1}{2}-|Z_{1}|n\ \geq\
(\lambda-4\delta_{3})\binom{n}{2}.\nolinebreak\mbox{\hskip
5.0pt\rule[-0.85pt]{2.0pt}{8.1pt}}\vskip 6.0pt plus 2.0pt minus 2.0pt$

Let $V^{\prime}:=V\setminus Z_{1}$, and let $H_{1}$ be the induced subgraph of
$H$ on $V^{\prime}$. Clearly, we have $|G_{1}\bigtriangleup
H_{1}|\leq|G\bigtriangleup H|\leq\delta_{2}{n\choose 3}$.

Define

$\displaystyle Z_{2}:=\left\\{v\in V^{\prime}\colon|L_{G_{1}}(v)\cap
L_{H_{1}}(v)|\leq(\lambda-\delta_{4})\binom{n}{2}\right\\}\quad\text{and}\quad
G_{2}:=G_{1}-Z_{2}.$

Similarly to Claim 34.1, we have the following.

###### Claim 34.2

We have $|Z_{2}|\leq\delta_{4}n$,
$|G_{2}|\geq\lambda\binom{n}{3}-\delta_{4}n^{3}$ and
$\delta(G_{2})\geq(\lambda-4\delta_{4})\binom{n}{2}$.

Let $V^{\prime\prime}:=V^{\prime}\setminus Z_{2}$ and $V_{{\bf
i}}^{\prime\prime}:=V_{{\bf i}}\cap V^{\prime\prime}$ for every legal sequence
${\bf i}$. Let $H_{2}$ be the induced subgraph of $H$ on $V^{\prime\prime}$.
Similarly to above, we have $|G_{2}\bigtriangleup H_{2}|\leq|G\bigtriangleup
H|\leq\delta_{2}n^{3}$ and

$\delta(H_{2})\geq(\lambda-4\delta_{2}){n\choose 2}.$ (43)

Take any $j\in S_{i}$. Our next aim is to show that $\partial
G\cap\binom{V_{j}^{\prime\prime}}{2}=\emptyset$. Suppose to the contrary that
there exists $D:=\\{u_{1},u_{2},u_{3}\\}\in G$ with $u_{1},u_{2}\in
V_{j}^{\prime\prime}$.

Let $H_{3}$ be obtained from $H_{2}$ by removing edges contained in
$H_{2}[V^{\prime\prime}_{{\bf i}}]$ for all legal sequences ${\bf i}$ of
length at least $\ell_{0}$. Let ${\bf T}:={\bf T}_{H_{3}}$. By the min-degree
property of $H$ and Lemma 19(c), each part of ${\bf V}_{H}$ at level at most
$\ell_{0}$ has at least $(\beta/2)^{\ell_{0}}n$ vertices. This is much larger
than $|Z_{1}\cup Z_{2}|\leq(\delta_{3}+\delta_{2})n$, the number of the
removed vertices when building $H_{3}$ from $H$. Thus, in particular, ${\bf
T}_{H_{3}}={\bf T}_{H}^{\mathrm{level}\leq\ell_{0}}$. Note that if the tree
${\bf T}$ is extendible then it is maximal up to level $\ell_{0}$. Indeed,
each involved recursive part $V_{{\bf i}}$ of $H$ has at least
$(\beta/2)^{\ell_{0}}n$ vertices and thus must span some edges, for otherwise
the edge density of the $P_{I}$-mixing construction $H$, which is
$\rho(H)\geq\rho(G)-\delta_{2}\geq\lambda-\delta-\delta_{2}$, can be increased
by

$(\lambda-o(1)){|V_{{\bf i}}|\choose 3}/{n\choose
3}\geq\lambda(\beta/2)^{3\ell_{0}}-o(1)\gg\delta+\delta_{2},$

thus jumping above the maximum density $\lambda+o(1)$, a contradiction. Let
$F$ be the rigid $P_{I}$-mixing construction with ${\bf T}_{F}={\bf T}$,
returned by Lemma 24 or 23. Since $M\gg\ell_{0}$, we can assume that $F$ has
at most $M-1$ vertices.

By relabelling $V(F)$ we can assume that $u_{1},u_{2}\in V(F)$ and these
vertices have the same branch in $F$ and $H$ (which is just the single-element
sequence $(j)$) while $u_{3}\not\in V(F)$. Let $F^{D}$ be obtained from $F$ by
adding the edge $D$ (and the new vertex $u_{3}$). Let ${\bf W}$ be the
partition structure of $F$.

###### Claim 34.3

$F^{D}\in\mathcal{F}_{M}$

Proof of Claim. Suppose to the contrary that there exists an embedding $f$ of
$F^{D}$ into some $P_{I}$-mixing construction $Q$ with base $P_{t}$ and bottom
partition $U_{1},\dots,U_{m_{t}}$. We can assume that $f(V(F^{D}))$ intersects
at least two parts $U_{s}$. First, suppose that already the image of
$V(F)\subseteq V(F^{D})$ under $f$ intersects at least two bottom parts of
$Q$. By the rigidity of $F$, we have that $t=i$ and, by relabelling the parts
$U_{s}$, we can assume that $W_{s}\subseteq U_{s}$ for every $s\in[m_{i}]$.
Thus $f(\\{u_{1},u_{2}\\})\subseteq U_{j}$. By $j\in S_{i}$, no edge of $Q$
can cover the pair $\\{u_{1},u_{2}\\}$, a contradiction to $f$ mapping $D$ to
an edge of $Q$. Thus, suppose that $f(V(F))\subseteq U_{s}$ for some
$s\in[m_{t}]$. Since $|F|>0$, we have $s\in R_{t}$. By
$V(F^{D})=V(F)\cup\\{u_{3}\\}$, it holds that $f(u_{3})\in U_{k}$ for some
$k\not=s$. However, then the profile of the edge $f(D)$ in $Q$ is
$\\{\\{\hskip 1.00006pt{s}^{(2)},k\hskip 1.00006pt\\}\\}$, contradicting by
Lemma 8 our assumption that $\lambda_{P_{t}}=\lambda<1$.

Now, we apply the familiar argument where we consider all maps $f:V(F^{D})\to
V(H_{3})$ such that $f$ is the identity on $D$ and, for every other vertex $u$
of $F^{D}$, it holds that $\mathrm{br}_{F}(u)=\mathrm{br}_{H_{3}}(f(u))$, that
is, the branch of $f(u)$ in $H_{3}$ is the same as the branch of $u$ in $F$.
We have by Lemma 19(c) that for every legal sequence ${\bf i}$ of length at
most $\ell_{0}$ we have $|V^{\prime\prime}_{{\bf
i}}|\geq(\beta/2)^{\ell_{0}}n$. Thus the number of choices of $f$ is at least,
say, $((\beta/2)^{\ell_{0}}n/2)^{v(F)-2}$. By Claim 34.3, each choice of $f$
reveals a missing edge $Y\in H_{3}\setminus G_{2}$. It is impossible that $Y$
is disjoint from $\\{u_{1},u_{2}\\}$ for at least half of the choices, for
otherwise the size of $H_{3}\setminus G_{2}$ is too large, since every such
$Y$ is overcounted at most $n^{v(F)-5}$ times. Thus at least half of the time
we have $Y\cap\\{u_{1},u_{2}\\}\not=\emptyset$. By $j\in S_{i}$, each such $Y$
intersects $\\{u_{1},u_{2}\\}$ in exactly one vertex. Note that, for each
$j\in\\{1,2\\}$, we have

$\displaystyle|L_{H_{3}}(u_{j})\setminus
L_{G_{2}}(u_{j})|\leq\Delta(H_{3})-\delta(G_{2})$
$\displaystyle\leq\Delta(H)-\delta(G_{2})$
$\displaystyle\leq(\lambda+\delta_{3})\binom{n}{2}-\left(\lambda\binom{n}{2}-4\delta_{1}^{1/2}n^{2}\right)\leq
2\delta_{3}n^{2}$

Thus the total number of such choices of $f$ can be upper bounded by
$4\delta_{3}n^{2}$, the number of choices of $Y$, times the trivial upper
bound $n^{v(F)-4}$ on the number of extensions to the remaining vertices of
$V(F)$. This contradicts that $\delta_{3}$ was sufficiently small depending on
$\beta$ and $\ell_{0}$.

We have shown that, for every $j\in S_{i}$, the set $V_{j}^{\prime\prime}$
spans no edges in $\partial G$, that is, every edge of $\partial G$ in $V_{j}$
must contain at least one vertex from $Z_{1}\cup Z_{2}$. Thus $\sum_{j\in
S_{i}}|(\partial G)[V_{j}]|\leq|Z_{1}\cup Z_{2}|n\leq\epsilon{n-1\choose 2}$.
This proves Lemma 34.

Now we are ready to prove Theorem 5.

Proof of Theorem 5. Again, we can assume that the assumptions and terminology
of Section 3.1 apply to $P_{I}$.

Let $M$ be a sufficiently large integer, in particular, such that $M$
satisfies Lemma 34 and the family ${\cal F}_{M}$ satisfies Theorem 2. Let us
show that ${\cal F}:={\cal F}_{M}$ satisfies Theorem 5. Theorem 2 implies that
$\pi({\cal F}_{M})=\pi({\cal F}_{\infty})=\lambda$, from which it easily
follow that $M({\cal F}_{\infty})\subseteq M({\cal F}_{M})$. Let us show the
converse implication.

Fix any $x$ in $M(\mathcal{F}_{M})$ and $\varepsilon>0$. We have to
approximate the point $(x,\lambda)\in\Omega({\cal F}_{M})$ by an element of
$\Omega({\cal F}_{\infty})$ within $\varepsilon$ in, say, the supremum norm.
Let $\left(G_{n}\right)_{n=1}^{\infty}$ be a sequence of
$\mathcal{F}_{M}$-free $3$-graphs that realizes $(x,\lambda)$. Let
$v_{n}:=v(G_{n})$ for $n\geq 1$. By passing to a subsequence if necessary, we
can assume that the sequence $(v_{n})_{n\in{\mathbb{N}}}$ is strictly
increasing. Since $\lim_{n\to\infty}\rho(G_{n})=\pi(\mathcal{F}_{M})$, by
Lemma 34, there exists a $P_{I}$-mixing construction $H_{n}$ on $V(G_{n})$ for
each $n$ such that $\delta(H_{n})\geq(\lambda-o(1))\binom{v_{n}-1}{2}$ and
$|G_{n}\triangle H_{n}|=o(v_{n}^{3})$ as $n\to\infty$. Since $H_{n}$ is a
$P_{I}$-mixing construction and
$\lim_{n\to\infty}\rho(H_{n})=\lim_{n\to\infty}\rho(G_{n})=\pi(\mathcal{F}_{M})$,
we have $\lim_{n\to\infty}\rho(\partial H_{n})\in M({\cal F}_{\infty})$

Let $\ell$ be a sufficiently large integer. Define
$\delta_{\ell+1}:=(1-\lambda/(2r))^{\ell+1}/2$ and let
$\delta_{\ell}\gg\dots\gg\delta_{1}$ be sufficiently small positive constants,
chosen in this order. Let $n\to\infty$. Let ${\bf V}$ be the partition
structure of $H_{n}$. By choosing $n$ sufficiently large we can assume that
$\delta(H_{n})\geq(\lambda-\delta_{1})\binom{v_{n}-1}{2}$, $|G_{n}\triangle
H_{n}|\leq\delta_{1}{v_{n}\choose 3}$, and
$|G_{n}|\geq(\lambda-\delta_{1})\binom{v_{n}}{3}$.

We call edges in $\partial G_{n}\setminus\partial H_{n}$ bad shadows. For
every bad shadow $e\in\partial G_{n}\setminus\partial H_{n}$ we let
$\mathrm{br}_{{\bf V}}(e)$ be the unique maximal sequence ${\bf i}$ such that
$e\subseteq V_{{\bf i}}$, and call $\mathrm{br}_{{\bf V}}(e)$ the branch of
$e$. Since each part $V_{i}$ has size at least $(\beta/2)v_{n}\geq r$, every
pair in $\partial G_{n}\setminus\partial H_{n}$ lies inside some $V_{i}$ and
thus has non-empty branch. Every bad shadow with branch of length at least
$\ell+1$ lies inside some part $V_{{\bf i}}$ with $|{\bf i}|\geq\ell+1$, whose
size is at most $(1-\lambda/(2r))^{\ell+1}v_{n}$ by Lemma 19(c). By the Hand-
Shaking Lemma, the number of such pairs is at most
$\frac{1}{2}\,(1-\lambda/(2r))^{\ell+1}v_{n}\cdot
v_{n}=\delta_{\ell+1}v_{n}^{2}$.

Recall that each part of height at most $\ell$ in $H_{n}$ has size at least
$(\beta/2)^{\ell}v_{n}$ by Lemma 19(c). In particular, the collection of all
bad shadows whose branch has length $1$ is exactly the set $\bigcup_{j\in
S_{b(H_{n})}}\partial G_{n}[V_{j}]$, and by Lemma 34, we have that

$\sum_{j\in S_{b(H_{n})}}|\partial G_{n}[V_{j}]|\leq\delta_{2}v_{n}^{2}.$

Now, repeat the following for every recursive index $j_{1}\in R_{b(H_{n})}$.
Since $\delta(H_{n})\geq(\lambda-\delta_{1})\binom{v_{n}-1}{2}$, we have by
Lemma 19 that $|V_{j_{1}}|\geq\beta v_{n}/2$ and
$\delta(H_{n}[V_{j_{1}}])\geq(\lambda-\delta_{2})\binom{|V_{j_{1}}|-1}{2}$.
Since $|G_{n}[V_{j_{1}}]\triangle H_{n}[V_{j_{1}}]|\leq|G_{n}\triangle
H_{n}|\leq\delta_{1}{v_{n}\choose 3}$, we have

$|G_{n}[V_{j_{1}}]|\geq|H_{n}[V_{j_{1}}]|-\delta_{1}{v_{n}\choose
3}\geq(\lambda-2\delta_{2})\binom{|V_{j_{1}}|}{3}.$

So applying Lemma 34 to $H_{n}[V_{j_{1}}]$, we obtain that, for example,

$\sum_{j\in S_{b(H_{n}[V_{j_{1}}])}}|\partial
G_{n}[V_{j_{1},j}]|\leq\frac{1}{m}\,{\delta_{3}{v_{n}\choose 2}},$

where $m:=\max\\{m_{k}\colon k\in I\\}$. Therefore,

$\sum_{j_{1}\in R_{b(H_{n})}}\sum_{j\in S_{b(H_{n}[V_{j_{1}}])}}|\partial
G_{n}[V_{j_{1},j}]|\leq m\frac{\delta_{3}{v_{n}\choose
2}}{m}=\delta_{3}{v_{n}\choose 2},$

that is, the number of bad shadows whose branch has length $2$ is at most
$\delta_{3}{v_{n}\choose 2}$. Repeating this argument we can show that the the
number of bad shadows whose length of branch is $h\leq\ell$ is at most
$\delta_{h+1}{v_{n}\choose 2}$. Therefore, the total number of bad shadows is
bounded by

$\displaystyle\delta_{2}{v_{n}\choose 2}+\delta_{3}{v_{n}\choose
2}+\cdots+\delta_{\ell}{v_{n}\choose 2}+\delta_{\ell+1}{v_{n}\choose 2}\leq
2\delta_{\ell+1}{v_{n}\choose 2}.$

Therefore, $|\partial G_{n}\setminus\partial H_{n}|/{v_{n}\choose
2}<\varepsilon/2$. On the other hand, every pair $xy\in\partial H_{n}$ at
level at most $\ell$ is covered by at least $(\beta/2)^{\ell}v_{n}$ triples in
$H_{n}$. This observation and our previous estimate of the total number of
pairs at levels at least $\ell+1$ give that

$|\partial H_{n}\setminus\partial
G_{n}|\leq\delta_{\ell+1}v_{n}^{2}+\frac{3|G_{n}\bigtriangleup
H_{n}|}{(\beta/2)^{\ell}v_{n}}<\frac{\varepsilon}{2}\,{v_{n}\choose 2}.$

Since $\varepsilon>0$ was arbitrary, we conclude that $x\in M({\cal F}_{M})$.
This proves Theorem 5.

##  6 Proof of Corollary 6

In this section we prove Corollary 6 by applying Theorem 5 to two specific
patterns.

Consider the following two patterns $P_{1}=(5,K_{5}^{3},\\{1\\})$ and
$P_{2}=(7,B_{5,3},\\{1\\})$, where the $3$-graph $B_{5,3}$ was defined in
Section 2.4.

###### Lemma 35

The following statements hold.

  1. (a)

We have
$\lambda_{P_{1}}=\lambda_{P_{2}}=\lambda:=\frac{3\left(\sqrt{7}-2\right)}{4}\approx
0.484313$.

  2. (b)

For ${\bf x}\in{\mathbb{S}}_{5}^{*}$ with $x_{1}\geq\ldots\geq x_{5}$, we have
$\lambda_{K_{5}^{3}}({\bf x})+\lambda x_{1}^{3}=\lambda$ if and only if

$\displaystyle x_{1}=\frac{\sqrt{7}-2}{3},\quad\text{and}\quad
x_{2}=\cdots=x_{5}=\frac{5-\sqrt{7}}{12}.$ (44)

  3. (c)

For ${\bf y}\in{\mathbb{S}}_{7}^{*}$ with $y_{1}\geq\ldots\geq y_{7}$, we have
$\lambda_{B_{5,3}}({\bf y})+\lambda y_{1}^{3}=\lambda$ if and only if

$\displaystyle y_{1}=\frac{\sqrt{7}-2}{3},\quad
y_{2}=y_{3}=\frac{5-\sqrt{7}}{12},\quad\text{and}\quad
y_{4}=\cdots=y_{7}=\frac{5-\sqrt{7}}{24}.$ (45)

Remark. Parts (b) and (c) imply that $P_{1}$ and $P_{2}$ are minimal.

Proof. For ${\bf x}\in\mathbb{S}_{5}^{*}$ and ${\bf y}\in\mathbb{S}_{7}^{*}$
as in the lemma, let

$\displaystyle g_{1}({\bf x})=\frac{\lambda_{K_{5}^{3}}({\bf
x})}{1-x_{1}^{3}}\quad\text{and}\quad g_{2}({\bf
y})=\frac{\lambda_{B_{5,3}}({\bf y})}{1-y_{1}^{3}}.$

It follows from the AM-GM inequality that

$\displaystyle g_{1}({\bf x})=\frac{\lambda_{K_{5}^{3}}({\bf
x})}{1-x_{1}^{3}}$
$\displaystyle=\frac{6\left(x_{1}\sum_{ij\in\binom{[2,5]}{2}}x_{i}x_{j}+\sum_{ijk\in\binom{[2,5]}{3}}x_{i}x_{j}x_{k}\right)}{1-x_{1}^{3}}$
$\displaystyle\leq\frac{6\left(x_{1}\binom{4}{2}\left(\frac{1-x_{1}}{4}\right)^{2}+\binom{4}{3}\left(\frac{1-x_{1}}{4}\right)^{3}\right)}{1-x_{1}^{3}}$
$\displaystyle=\frac{3(1-x_{1})(1+5x_{1})}{8(1+x_{1}+x_{1}^{2})}\leq\frac{3(\sqrt{7}-2)}{4},$

where the equality holds if and only if (44) holds.

Similarly, for $g_{2}({\bf y})$ we have

$\displaystyle\lambda_{B_{5,3}}({\bf y})$
$\displaystyle=6y_{1}\left(y_{2}y_{3}+(y_{2}+y_{3})(y_{4}+y_{5}+y_{6}+y_{7})+(y_{4}+y_{5})(y_{6}+y_{7})\right)$
$\displaystyle\quad+6\left(y_{2}y_{3}(y_{4}+y_{5}+y_{6}+y_{7})+y_{2}(y_{4}+y_{6})(y_{5}+y_{7})+y_{3}(y_{4}+y_{7})(y_{5}+y_{6})\right).$

Notice that

$\displaystyle
y_{2}y_{3}+(y_{2}+y_{3})(y_{4}+y_{5}+y_{6}+y_{7})+(y_{4}+y_{5})(y_{6}+y_{7})$
$\displaystyle=y_{2}y_{3}+y_{2}(y_{4}+y_{5})+y_{2}(y_{6}+y_{7})+y_{3}(y_{4}+y_{5})+y_{3}(y_{6}+y_{7})+(y_{4}+y_{5})(y_{6}+y_{7})$
$\displaystyle=\sigma_{2}(y_{2},y_{3},y_{4}+y_{5},y_{6}+y_{7})\leq\binom{4}{2}\left(\frac{y_{2}+\cdots+y_{7}}{4}\right)^{2}=\frac{3}{8}\left(1-y_{1}\right)^{2},$

where

$\sigma_{k}(x_{1},\dots,x_{s}):=\sum_{X\subseteq[s]\atop|X|=k}\prod_{i\in
X}x_{i}$

is the $i$-th symmetric polynomial, and the last inequality follows from the
Maclaurin Inequality (see e.g. [4, Theorem 11.2]) that
$\sigma_{k}(x_{1},\dots,x_{s})\leq{s\choose k}(x_{1}+\dots+x_{s})^{s}/s^{k}$
for any non-negative $x_{i}$’s and $k\in{\mathbb{N}}$.

On the other hand, it follows from the AM-GM and Maclaurin inequalities that

$\displaystyle
y_{2}y_{3}(y_{4}+y_{5}+y_{6}+y_{7})+y_{2}(y_{4}+y_{6})(y_{5}+y_{7})+y_{3}(y_{4}+y_{7})(y_{5}+y_{6})$
$\displaystyle\leq
y_{2}y_{3}(y_{4}+y_{5}+y_{6}+y_{7})+y_{2}\left(\frac{y_{4}+y_{6}+y_{5}+y_{7}}{2}\right)^{2}+y_{3}\left(\frac{y_{4}+y_{7}+y_{5}+y_{6}}{2}\right)^{2}$
$\displaystyle=\sigma_{3}(y_{2},y_{3},(y_{4}+y_{5}+y_{6}+y_{7})/2,(y_{4}+y_{5}+y_{6}+y_{7})/2)\leq\binom{4}{3}\left(\frac{y_{2}+\cdots+y_{7}}{4}\right)^{3}=\frac{1}{4}(1-y_{1})^{3}.$

Therefore, $\lambda_{B_{5,3}}({\bf y})\leq
6\left(\frac{3}{8}y_{1}\left(1-y_{1}\right)^{2}+\frac{1}{4}(1-y_{1})^{3}\right)$.
Similar to $g_{1}({\bf x})$ we obtain

$\displaystyle g_{2}({\bf y})=\frac{\lambda_{B_{5,3}}({\bf
y})}{1-y_{1}^{3}}\leq\frac{6\left(\frac{3}{8}y_{1}\left(1-y_{1}\right)^{2}+\frac{1}{4}(1-y_{1})^{3}\right)}{1-y_{1}^{3}}\leq\frac{3(\sqrt{7}-2)}{4},$

and equality holds if and only if (45) holds. This gives all claims of the
lemma.

For the next lemma we need the following definitions. Given a collection
$\\{P_{i}=(m_{i},E_{i},R_{i})\colon i\in I\\}$ of $r$-uniform patterns, we
define a family $\Sigma^{k}P_{I}$ recursively for every integer $k\geq 0$ in
the following way. (Note that it is different from the family
$\Sigma_{k}P_{I}$ that appeared in the proof of Lemma 14.)

######  Definition 36 (The $k$-th $P_{I}$-mixing construction)

Let

$\displaystyle\Sigma^{0}P_{I}:=\bigcup_{i\in I}\left\\{H\colon\text{$H$ is a
$P_{i}$-construction}\right\\}.$

For every integer $k\geq 1$ an $r$-graph $H$ is a $k$-th $P_{I}$-mixing
construction if there exist $i\in I$ and a partition $V(H)=V_{1}\cup\cdots\cup
V_{m_{i}}$ such that $H$ can be obtained from the blowup
$E_{i}(\\!(V_{1},\ldots,V_{m_{i}})\\!)$ by adding, for each $j\in R_{i}$, an
arbitrary $(k-1)$-th $P_{I}$-mixing construction on $V_{j}$. Let
$\Sigma^{k}P_{I}$ denote the collection of all $k$-th $P_{I}$-mixing
constructions.

It easy to see that $\Sigma^{k}P_{I}\subseteq\Sigma^{k^{\prime}}P_{I}$ for all
$k^{\prime}\geq k$ and, if $\lambda_{P_{i}}=\lambda$ for each $i\in I$, then
the maximum asymptotic density attainable by $\Sigma^{k}P_{I}$ for each $k\geq
0$ is $\lambda$. Moreover,

$\displaystyle\bigcup_{k\geq 0}\Sigma^{k}P_{I}=\Sigma P_{I}.$

For every integer $k\geq 0$ let $M_{\Sigma^{k}P_{I}}$ (resp. $M_{\Sigma
P_{I}}$) be the collection of points $x\in[0,1]$ such that there exists a
sequence $\left(H_{n}\right)_{n=1}^{\infty}$ of $r$-graphs in
$\Sigma^{k}P_{I}$ (resp. $\Sigma P_{I}$) with

$\displaystyle\lim_{n\to\infty}v(H_{n})=\infty,\quad\lim_{n\to\infty}\rho(H_{n})=\lambda_{P_{I}},\quad\text{and}\quad\lim_{n\to\infty}\rho(\partial
H_{n})=x.$

It is easy to observe that the set $M_{\Sigma P_{I}}$ is the closure of
$\bigcup_{k\geq 0}M_{\Sigma^{k}P_{I}}$, i.e.

$\displaystyle M_{\Sigma P_{I}}=\overline{\bigcup_{k\geq
0}M_{\Sigma^{k}P_{I}}}.$

We will need the following theorem (which is a rather special case of e.g.
[16, Theorems (1) and (3)]) for determining the Hausdorff dimension of a self-
similar set.

###### Theorem 37

Suppose that and $m\geq 2$ and, for each $i\in[m]$, we have a linear map
$\phi_{i}(x):=r_{i}(x-x_{i})+x_{i}$ for some $r_{i},x_{i}\in{\mathbb{R}}$ with
$|r_{i}|<1$. Suppose that this collection satisfies the open set condition,
namely that there exists a non-empty open set $V$ such that

  1. 1.

$\bigcup_{i\in[m]}\psi_{i}(V)\subseteq V$, and

  2. 2.

the sets in $\\{\psi_{i}(V)\colon i\in[m]\\}$ are pairwise disjoint.

Then the Hausdorff dimension $d$ of the (unique) bounded closed non-empty set
$A$ that satisfies $A=\bigcup_{i\in[m]}\psi_{i}(A)$ is the (unique) solution
of the equation

$\displaystyle\sum_{i\in[m]}|r_{i}|^{d}=1.$

The following lemma determines the Hausdorff dimension of the set
$M_{\Sigma\\{P_{1},P_{2}\\}}$.

###### Lemma 38

We have
$M_{\Sigma^{0}\\{P_{1},P_{2}\\}}=\left\\{\frac{6-\sqrt{7}}{4},\frac{22-3\sqrt{7}}{16}\right\\}$,
and for every integer $k\geq 0$ we have

$\displaystyle M_{\Sigma^{k+1}\\{P_{1},P_{2}\\}}=\left\\{\psi_{1}(x)\colon
x\in M_{\Sigma^{k}\\{P_{1},P_{2}\\}}\right\\}\cup\left\\{\psi_{2}(x)\colon
x\in M_{\Sigma^{k}\\{P_{1},P_{2}\\}}\right\\},$ (46)

where we define, for $x\in{\mathbb{R}}$,

$\displaystyle\psi_{1}(x):=\frac{13\sqrt{7}-20}{18}+\left(\frac{\sqrt{7}-2}{3}\right)^{2}x\quad\text{and}\quad\psi_{2}(x):=\frac{47\sqrt{7}-64}{72}+\left(\frac{\sqrt{7}-2}{3}\right)^{2}x.$

Moreover, the Hausdorff dimension of $M_{\Sigma\\{P_{1},P_{2}\\}}$ is
$\delta:=\frac{\log 2}{\log\left(4\sqrt{7}+11\right)}\approx 0.225641$.

Proof. Lemma 35 implies that $M_{\Sigma^{0}{\\{P_{1},P_{2}\\}}}=\\{a,b\\}$,
where $a:=\frac{6-\sqrt{7}}{4}$ and $b:=\frac{22-3\sqrt{7}}{16}$. Now,
equation (46) follows from the definition of
$M_{\Sigma^{k+1}\\{P_{1},P_{2}\\}}$ and Lemma 35. So it suffices to prove that
the Hausdorff dimension of $M_{\Sigma\\{P_{1},P_{2}\\}}$ is $\delta$. Let
$A:=M_{\Sigma\\{P_{1},P_{2}\\}}$ and $V:=(a,b)$. Note that note that $A$ is a
bounded closed set that by (46), satisfies $A=\psi_{1}(A)\cup\psi_{2}(A)$.
Also, routine calculations show that $\psi_{1}(V)=(a,c)$ and
$\psi_{2}(V)=(d,b)$, where $c:=\frac{166-17\sqrt{7}}{144}$ is less than
$d=\frac{124-23\sqrt{7}}{72}$. Thus the open set $V$ and the maps $\psi_{1}$
and $\psi_{2}$ satisfy the Open Set Condition, i.e.

$\displaystyle\psi_{1}(V)\cup\psi_{2}(V)\subseteq
V\quad\text{and}\quad\psi_{1}(V)\cap\psi_{2}(V)=\emptyset.$

On the other hand, it is easy to check that $A\subseteq[0,1]$ is a closed set
satisfying $A=\psi_{1}(A)\cup\psi_{2}(A)$. By Theorem 37, the Hausdorff
dimension of $A$ is the unique solution to

$\displaystyle\left(\frac{\sqrt{7}-2}{3}\right)^{2x}+\left(\frac{\sqrt{7}-2}{3}\right)^{2x}=1,$

which is $\frac{\log 2}{\log\left(4\sqrt{7}+11\right)}$.

Now Corollary 6 is an easy consequence of Theorem 5 and Lemma 38.

##  7 Concluding remarks

Since our paper is quite long, we restricted applications of Theorem 2 to
$3$-graphs only. Some of these results extends to general $r$ while such an
extension for others seems quite challenging. For example, our proof of
Theorem 5 also works for $r$-uniform patterns and the $(r-2)$-th shadow (when
we consider pairs of vertices covered by edges). However, we do not know if
the analogue of Theorem 5 holds for the $(r-i)$-th shadow when $i\geq 3$.

On the other hand, we tried to present a fairly general version of Theorem 2
(for example, allowing $P_{I}$ to contain patterns with different Lagrangians)
in case it may be useful for some other applications.

In some rather special cases of Theorem 2, it may be possible to drop the
constraint that each $P_{i}\in P$ is minimal. For example, it is shown in [15]
that for every (not necessarily minimal) pattern $(m,H,\emptyset$) where $H$
consists of simple $r$-sets, there exists a finite forbidden family whose
extremal Turán constructions are exactly maximum blowups of $H$. However, we
do not know if this is true in general, since the minimality condition is
crucially used in the proof of Theorem 2.

## Acknowledgements

We would like to thank József Balogh, Felix Christian Clemen, Victor Falgas-
Ravry, and Dhruv Mubayi for helpful discussions and/or comments.

## References

  * [1] R. Baber and J. Talbot.  Hypergraphs do jump.  Combin. Probab. Comput., 20(2):161–171, 2011.
  * [2] J. Balogh, F. C. Clemen, and H. Luo.  Non-degenerate hypergraphs with exponentially many extremal constructions.  E-print arxiv:2208.00652, 2022.
  * [3] W. G. Brown.  On an open problem of Paul Turán concerning 3-graphs.  In Studies in pure mathematics, pages 91–93. Springer, 1983.
  * [4] Z. Cvetkovski.  Inequalities.  Springer, Heidelberg, 2012.  Theorems, techniques and selected problems.
  * [5] P. Erdős.  On extremal problems of graphs and generalized graphs.  Israel J. Math., 2:183–190, 1964.
  * [6] P. Erdős.  Some recent results on extremal problems in graph theory. Results.  In Theory of Graphs (Internat. Sympos., Rome, 1966), pages 117–123 (English); pp. 124–130 (French). Gordon and Breach, New York, 1967.
  * [7] P. Erdős.  On the combinatorial problems I would most like to see solved.  Combinatorica, 1:25–42, 1981.
  * [8] P. Erdős and M. Simonovits.  A limit theorem in graph theory.  Stud. Sci. Math. Hungar., 1:51–57, 1966.
  * [9] P. Erdős and A. H. Stone.  On the structure of linear graphs.  Bull. Amer. Math. Soc., 52:1087–1091, 1946.
  * [10] D. G. Fon-der Flaass.  A method for constructing $(3,4)$-graphs.  Mat. Zametki, 44:546–550, 1988.
  * [11] P. Frankl and Z. Füredi.  An exact result for 3-graphs.  Discrete Math., 50:323–328, 1984.
  * [12] P. Frankl, Y. Peng, V. Rödl, and J. Talbot.  A note on the jumping constant conjecture of Erdős.  J. Combin. Theory (B), 97:204–216, 2007.
  * [13] P. Frankl and V. Rödl.  Hypergraphs do not jump.  Combinatorica, 4(2-3):149–159, 1984.
  * [14] A. Frohmader.  More constructions for Turán’s $(3,4)$-conjecture.  Electron. J. Combin., 15(1):Research Paper 137, 23, 2008.
  * [15] J. Hou, H. Li, X. Liu, D. Mubayi, and Y. Zhang.  Hypergraphs with infinitely many extremal constructions.  arXiv preprint arXiv:2206.03948, 2022.
  * [16] J. E. Hutchinson.  Fractals and self-similarity.  Indiana Univ. Math. J., 30(5):713–747, 1981.
  * [17] G. Katona.  A theorem of finite sets.  In Theory of graphs (Proc. Colloq., Tihany, 1966), pages 187–207, 1968.
  * [18] G. O. H. Katona, T. Nemetz, and M. Simonovits.  On a graph problem of Turán (In Hungarian).  Mat. Fiz. Lapok, 15:228–238, 1964.
  * [19] P. Keevash.  Hypergraph Turán problems.  In Surveys in combinatorics 2011, volume 392 of London Math. Soc. Lecture Note Ser., pages 83–139. Cambridge Univ. Press, Cambridge, 2011.
  * [20] P. Keevash.  Counting designs.  J. Europ. Math. Soc, 20:903–927, 2018.
  * [21] T. P. Kirkman.  On a problem in combinations.  Cambridge and Dublin Math. J., 2:191–204, 1847.
  * [22] A. V. Kostochka.  A class of constructions for Turán’s (3,4)-problem.  Combinatorica, 2:187–192, 1982.
  * [23] J. B. Kruskal.  The number of simplices in a complex.  In Mathematical optimization techniques, pages 251–278. Univ. of California Press, Berkeley, Calif., 1963.
  * [24] X. Liu.  Cancellative hypergraphs and steiner triple systems.  arXiv preprint arXiv:1912.11917, 2019.
  * [25] X. Liu and D. Mubayi.  The feasible region of hypergraphs.  J. Comb. Theory, Ser. B, 148:23–59, 2021.
  * [26] X. Liu and D. Mubayi.  A hypergraph Turán problem with no stability.  Combinatorica, pages 1–30, 2022.
  * [27] X. Liu, D. Mubayi, and C. Reiher.  Hypergraphs with many extremal configurations.  arXiv preprint arXiv:2102.02103, 2021.
  * [28] X. Liu, D. Mubayi, and C. Reiher.  Hypergraphs with many extremal configurations II.  In preparation, 2022.
  * [29] T. S. Motzkin and E. G. Straus.  Maxima for graphs and a new proof of a theorem of Turán.  Canadian J. Math., 17:533–540, 1965.
  * [30] S. Norin and L. Yepremyan.  Turán number of generalized triangles.  J. Combin. Theory (A), 146:312–343, 2017.
  * [31] O. Pikhurko.  On possible Turán densities.  Israel J. Math., 201(1):415–454, 2014.
  * [32] O. Pikhurko.  The maximal length of a gap between $r$-graph Turán densities.  Electronic J. Combin., 22:7pp., 2015.
  * [33] O. Pikhurko, J. Sliacǎn, and K. Tyros.  Strong forms of stability from flag algebra calculations.  J. Combin. Theory (B), 135:129–178, 2019.
  * [34] V. Rödl and M. Schacht.  Generalizations of the removal lemma.  Combinatorica, 29(4):467–501, 2009.
  * [35] A. Sidorenko.  What we know and what we do not know about Turán numbers.  Graphs Combin., 11(2):179–199, 1995.
  * [36] M. Simonovits.  A method for solving extremal problems in graph theory, stability problems.  In Theory of Graphs (Proc. Colloq., Tihany, 1966), pages 279–319. Academic Press, New York, 1968.
  * [37] P. Turán.  On an extremal problem in graph theory (in Hungarian).  Mat. Fiz. Lapok, 48:436–452, 1941.
  * [38] Z. Yan and Y. Peng.  Non-jumping Turán densities of hypergraphs.  Discrete Math., 346(1):Paper No. 113195, 11, 2023.


# Retrolensing by light rays slightly inside and outside of a photon sphere
around a Reissner-Nordström naked singularity

Naoki Tsukamoto1<EMAIL_ADDRESS>1Department of General Science and
Education, National Institute of Technology, Hachinohe College, Aomori
039-1192, Japan

###### Abstract

We investigate retrolensing of sun lights reflected by a photon sphere around
a Reissner-Nordström naked singularity. We apply the deflection angles of
light rays reflected by slightly outside the photon sphere and rays reflected
by a potential barrier near an antiphoton sphere in the strong deflection
limits to the retrolensing. We show that the retrolensing by the photon sphere
around the Reissner-Nordström naked singularity can be brighter than the one
around a Reissner-Nordström black hole because of the existence of the rays
which reflected by the potential barrier.

##  I Introduction

Gravitational lensing in a weak field gravitational with a small deflection
angle $\alpha\ll 1$ has been investigated intensively for finding dark and
massive objects such as dark matters and extrasolar planets
Schneider_Ehlers_Falco_1992 ; Schneider_Kochanek_Wambsganss_2006 . On the
other hand, the images of light rays deflected in a strong gravitational field
caused by a compact object have also been considered intermittently. In 1931
Hagihara_1931  Hagihara pointed out that an observer can see light rays
reflected by a photon sphere which is a sphere made of unstable circular light
orbits and Darwin Darwin_1959  and several authors revisited the images
reflected by the photon sphere Atkinson_1965 ; Luminet_1979 ; Ohanian_1987 ;
Nemiroff_1993 ; Frittelli_Kling_Newman_2000 ; Virbhadra_Ellis_2000 ;
Bozza_Capozziello_Iovane_Scarpetta_2001 ; Bozza:2002zj ; Perlick:2003vg ;
Virbhadra:2008ws ; Bozza_2010 ; Tsukamoto:2016zdu ; Shaikh:2019jfr ;
Shaikh:2019itn ; Tsukamoto:2020uay ; Tsukamoto:2020iez ; Paul:2020ufc .
Recently, gravitational waves emitted by black hole binaries have been
reported by LIGO and Virgo Collaborations Abbott:2016blz  and a shadow at the
center of a giant elliptical galaxy M87 has been detected by Event Horizon
Telescope Collaboration Akiyama:2019cqa . Gravitational lensing with the
deflection angle $\alpha>1$ in the strong gravitational field and other
phenomena will be more significant before.

It is known that there is a sphere of unstable (stable) circular light orbits
called photon sphere (antiphoton sphere) around compact objects such as black
holes Perlick_2004_Living_Rev ; Claudel:2000yi ; Perlick:2021aok  and their
theoretical and observational aspects were investigated eagerly Hod:2017xkz ;
Sanchez:1977si ; Press:1971wr ; Abramowicz_Prasanna_1990 ; Mach:2013gia ;
Barcelo:2000ta ; Ames_1968 . The generalizations or alternative surfaces of
the photon sphere have been also suggested by several researchers
Claudel:2000yi ; Gibbons:2016isj . It is concerned that the antiphoton sphere
may lead to instability of ultra compact objects because of the slow decay of
linear waves Keir:2014oka ; Cardoso:2014sna ; Cunha:2017qtt .

In 2002, Bozza Bozza:2002zj  has investigated gravitational lensing of light
rays reflected by a photon sphere in a general asymptotically-flat, static and
spherical symmetric spacetime in a strong deflection limit $b\rightarrow
b_{\mathrm{m}}+0$, where $b$ is the impact parameter of a light and
$b_{\mathrm{m}}$ is its critical impact parameter. The deflection angle of the
light ray is expressed by

$\displaystyle\alpha$ $\displaystyle=$
$\displaystyle-\bar{a}\log\left(\frac{b}{b_{\mathrm{m}}}-1\right)+\bar{b}$ (1)
$\displaystyle+O\left(\left(\frac{b}{b_{\mathrm{m}}}-1\right)\log\left(\frac{b}{b_{\mathrm{m}}}-1\right)\right),$

where $\bar{a}$ and $\bar{b}$ are determined by the metric of the spacetime.
111In Ref. Bozza:2002zj , the order of the vanishing term in the deflection
angle in the strong deflection limit is considered as $O(b-b_{\mathrm{m}})$
but we should read the order as
$O\left(\left(b/b_{\mathrm{m}}-1\right)\log\left(b/b_{\mathrm{m}}-1\right)\right)$
as shown in Refs. Iyer:2006cn ; Tsukamoto:2016qro ; Tsukamoto:2016jzh .  The
gravitational lensing in strong deflection limit has been investigated in many
aspects and the details of its analysis has been studied Bozza:2002af ;
Eiroa:2002mk ; Petters:2002fa ; Eiroa:2003jf ; Bozza:2004kq ; Bozza:2005tg ;
Bozza:2006sn ; Bozza:2006nm ; Iyer:2006cn ; Bozza:2007gt ; Tsukamoto:2016qro ;
Ishihara:2016sfv ; Tsukamoto:2016oca ; Tsukamoto:2016zdu ; Tsukamoto:2016jzh ;
Tsukamoto:2017edq ; Shaikh:2019jfr ; Shaikh:2019itn ; Tsukamoto:2020uay ;
Tsukamoto:2020iez ; Paul:2020ufc ; Hsieh:2021scb ; Aldi:2016ntn ;
Takizawa:2021gdp ; Tsukamoto:2020bjm ; Tsukamoto:2021caq ; Aratore:2021usi .

In 2019, Shaikh et al. Shaikh:2019itn  have investigated gravitational lensing
of lays which pass a photon sphere and which are reflected by a potential
barrier near an antiphoton sphere in a general asymptotically-flat, static and
spherical symmetric spacetime without an event horizon. The deflection angle
in a strong deflection limit $b\rightarrow b_{\mathrm{m}}-0$ is expressed by

$\displaystyle\alpha$ $\displaystyle=$
$\displaystyle-\bar{c}\log\left(\frac{b_{\mathrm{m}}}{b}-1\right)+\bar{d}$ (2)
$\displaystyle+O\left(\left(\frac{b_{\mathrm{m}}}{b}-1\right)\log\left(\frac{b_{\mathrm{m}}}{b}-1\right)\right),$

where $\bar{c}$ and $\bar{d}$ are calculated from the metric of the spacetime.

Compact objects in nature would not have a large amount of an electric charge
since charged objects will be neutralized quickly. However, in general
relativity, the Reissner-Nordström spacetime is often considered as a toy
model of a compact object since its property would be similar to other
spacetimes such as a Hayward spacetime Chiba:2017nml  and since we might treat
it analytically. A shadow deVries:2000 ; Takahashi:2005hy ; Zakharov:2014lqa ;
Akiyama:2019cqa ; Akiyama:2019eap ; Kocherlakota:2021dcv , time delay of rays
Sereno:2003nd , gravitational lensing Bin-Nun:2010exl ; Bin-Nun:2010lws  in
the Reissner-Nordström spacetime have been investigated.

Eiroa et al. have considered the deflection angle in the strong defection
limit $r_{0}\rightarrow r_{\mathrm{m}}+0$, where $r_{0}$ and $r_{m}$ are the
radial coordinate of the closest position of a light ray and the position of
the photon sphere, respectively, or $b\rightarrow b_{\mathrm{m}}+0$
numerically in the Reissner-Nordström spacetime Eiroa:2002mk . Bozza
Bozza:2002zj  has obtained the analytical form of $\bar{a}$ in Eq. (1) and
$\bar{b}$ was partly calculated in numerical in the strong deflection limit
$b\rightarrow b_{\mathrm{m}}+0$. The exact analytical expressions of $\bar{a}$
and $\bar{b}$ has been obtained by Tsukamoto Tsukamoto:2016jzh ;
Tsukamoto:2016oca . The gravitational lensing in the strong deflection limit
$b\rightarrow b_{\mathrm{m}}+0$ by a marginal unstable photon sphere has also
been considered by Tsukamoto Tsukamoto:2020iez . Gravitational lensing by a
photon sphere around a singularity in the a Reissner-Nordström spacetime with
$q^{2}/m^{2}=1.05$ in the strong defection limit $b\rightarrow
b_{\mathrm{m}}-0$ has been investigated by Shaikh et al. Shaikh:2019itn  and
exact analytical form of $\bar{c}$ and $\bar{d}$ in Eq. (2) has been
calculated by Tsukamoto Tsukamoto:2021fsz .

Holz and Wheeler Holz:2002uf  have investigated retrolensing with the
deflection angle $\alpha\sim\pi$ in a lens configuration that the sun as the
source of a light, an observer, and a black hole as a lens object line up in
this order. Retrolensing by a photon sphere around a black hole
DePaolis:2003ad ; Eiroa:2003jf ; Bozza:2004kq ; DePaolis:2004xe ;
Abdujabbarov:2017pfw , around a wormhole Tsukamoto:2017edq ; Tsukamoto:2016zdu
, and a naked singularity ZamanBabar:2021zuk  have been investigated.
Retrolensing with a deflection angle $\alpha\sim 3\pi$ also have been studied
Tsukamoto:2017edq . Retrolensing by a Reissner-Nordström black hole in our
galaxy Eiroa:2003jf  and near our solar system Tsukamoto:2016oca  have been
studied.

In this paper, we apply the retrolensing of light rays reflected by a photon
sphere around a Reissner-Nordström naked singularity by using the exact
analytic expressions of $\bar{a}$ and $\bar{b}$ in Eq. (1) in the strong
deflection limit $b\rightarrow b_{\mathrm{m}}+0$ and $\bar{c}$ and $\bar{d}$
in Eq. (2) in the strong deflection limit $b\rightarrow b_{\mathrm{m}}-0$. The
retrolensing by the photon sphere around the Reissner-Nordström naked
singularity can be brighter than the one around a Reissner-Nordström black
hole because of the existence of the rays which reflected by a potential
barrier near an antiphoton sphere.

This paper is organized as follows. We review the deflection angle of rays in
the strong deflection limits in the Reissner-Nordström spacetime in Sec. II,
we investigate the retrolensing by the photon sphere around the Reissner-
Nordström naked singularity in Sec. III, and we discuss and conclude our
results in Sec. IV. In this paper, we use the units in which the light speed
and Newton’s constant are unity.

##  II Deflection angle in the Reissner-Nordström spacetime

In this section, we review shortly the deflection angle of light rays in
strong deflection limits in a Reissner-Nordström spacetime. See Refs.
Tsukamoto:2016jzh ; Tsukamoto:2016oca ; Tsukamoto:2021fsz  for its detail. The
Reissner-Nordström spacetime has a line element

$ds^{2}=-A(r)dt^{2}+\frac{dr^{2}}{A(r)}+r^{2}(d\vartheta^{2}+\sin^{2}\vartheta
d\varphi^{2}),$ (3)

where $A(r)$ is given by

$A(r)=1-\frac{2m}{r}+\frac{q^{2}}{r^{2}}$ (4)

and $m\geq 0$ is an Arnowitt-Deser-Misner (ADM) mass and $q\geq 0$ is an
electrical charge and it has time-translational and axial Killing vectors
$t^{\mu}\partial_{\mu}=\partial_{t}$ and
$\varphi^{\mu}\partial_{\mu}=\partial_{\varphi}$ because of its stationarity
and axisymmetry, respectively. There is an event horizon $r=r_{\mathrm{H}}$,
where

$r_{\mathrm{H}}\equiv m+\sqrt{m^{2}-q^{2}},$ (5)

for $q\leq m$ while the spacetime has a naked singularity for $m<q$. Without
loss of generality, we can assume $\vartheta=\pi/2$ because of spherical
symmetry.

By using a wave vector $k^{\mu}\equiv\dot{x}^{\mu}$, where the dot denotes a
differentiation with respect to an affine parameter, the trajectory of a light
ray is expressed by $k^{\mu}k_{\mu}=0$ and it can be rewritten in

$-A\dot{t}^{2}+\frac{\dot{r}^{2}}{A}+r^{2}\dot{\varphi}^{2}=0.$ (6)

The light ray can be characterized by its closest distant or its reflectional
point $r=r_{0}$ and an equation

$A_{0}\dot{t}^{2}_{0}=r^{2}_{0}\dot{\varphi}^{2}_{0}$ (7)

holds at the reflectional point. Here and hereinafter functions with subscript
$0$ denote the function at the reflectional point $r=r_{0}$. The impact
parameter of a light is given by

$b(r_{0})\equiv\frac{L}{E}=\frac{r_{0}^{2}\dot{\varphi}_{0}}{A_{0}\dot{t}_{0}},$
(8)

where $E\equiv-g_{\mu\nu}t^{\mu}k^{\nu}=A\dot{t}$ and $L\equiv
g_{\mu\nu}\varphi^{\mu}k^{\nu}=r^{2}\dot{\varphi}$ are conserved energy and
conserved angular momentum of the ray, respectively. From Eq. (7), the impact
parameter can be expressed by

$b=\pm\sqrt{\frac{r_{0}^{2}}{A_{0}}}.$ (9)

From Eq. (6), the radial motion of the light is given by

$\dot{r}^{2}+\frac{V(r)}{E^{2}}=0,$ (10)

where $V(r)$ is an effective potential is defined by

$V(r)\equiv\frac{Ab^{2}}{r^{2}}-1.$ (11)

The ray can exist in a region $V(r)\leq 0$. There are a photon sphere at
$r=r_{\mathrm{m}}$, where $r_{\mathrm{m}}$ is given by

$r_{\mathrm{m}}=\frac{3m+\sqrt{9m^{2}-8q^{2}}}{2}$ (12)

for $0\leq q<3m/(2\sqrt{2})$ and an antiphoton sphere at $r=r_{\mathrm{aps}}$,
where $r_{\mathrm{aps}}$ is

$r_{\mathrm{aps}}=\frac{3m-\sqrt{9m^{2}-8q^{2}}}{2}$ (13)

for $m<q<3m/(2\sqrt{2})$. Note that
$V(r_{\mathrm{m}})=V^{\prime}(r_{\mathrm{m}})=0$ and
$V^{\prime\prime}(r_{\mathrm{m}})<0$ holds for a critical impact parameter
$b_{\mathrm{m}}\equiv b(r_{0}=r_{\mathrm{m}})$ and that
$V(r_{\mathrm{aps}})=V^{\prime}(r_{\mathrm{aps}})=0$ and
$V^{\prime\prime}(r_{\mathrm{aps}})>0$ hold for a impact parameter. Here, the
prime is the differentiation with respect to the radial coordinate $r$. We
define the smaller positive zero point $r=r_{c}$ of the effective potential of
the light ray with impact critical parameter $b_{\mathrm{m}}$ for
$m<q<3m/(2\sqrt{2})$. We obtain $r_{\mathrm{c}}$ as

$r_{\mathrm{c}}=\frac{r_{\mathrm{m}}\left(\sqrt{mr_{\mathrm{m}}}-\sqrt{mr_{\mathrm{m}}-q^{2}}\right)}{\sqrt{mr_{\mathrm{m}}-q^{2}}}.$
(14)

We note that $b(r_{0}=r_{\mathrm{c}})=b_{\mathrm{m}}$. We plot the radial
coordinates of the photon sphere $r_{\mathrm{m}}$, an antiphoton sphere
$r_{\mathrm{aps}}$, an event horizon $r_{\mathrm{H}}$, and the smaller
positive zero point $r_{\mathrm{c}}$ of effective potential in Fig. 1.

Figure 1: The reduced radial coordinates of a photon sphere
$r_{\mathrm{m}}/m$, an antiphoton sphere $r_{\mathrm{aps}}/m$, an event
horizon $r_{\mathrm{H}}/m$, and the smaller positive zero point of effective
potential $r_{\mathrm{c}}/m$. A (red) dot-dashed, a (green) dashed, a (blue)
dotted, and a (black) solid curves denote $r_{\mathrm{m}}/m$,
$r_{\mathrm{aps}}/m$, $r_{\mathrm{H}}/m$, and $r_{\mathrm{c}}/m$,
respectively.

We consider that a light ray comes from at a spatial infinity and it is
reflected at $r=r_{0}$. From Eq. (6), the deflection angle $\alpha$ of the ray
is obtained as

$\alpha=2\int^{\infty}_{r_{0}}\frac{dr}{r\sqrt{\frac{r^{2}}{b^{2}}-A}}-\pi.$
(15)

###  II.1 Light rays slightly outside of the photon sphere

We consider that light rays are reflected nearly outside of the photon sphere.
The examples of the effective potentials for $q\leq m$ and
$m<q<3m/(2\sqrt{2})$ are shown in the left panel and middle panel,
respectively, in Fig. 2.

Figure 2: Effective potentials $V$. Dashed (green) curves denotes $V$ with
$b=b_{\mathrm{m}}$. A left panel shows a black hole case with $q/m=0.9$ and a
solid (red) curve denote $V$ with $b=1.01b_{\mathrm{m}}/m$. Middle and right
panels show a naked singularity case with $q/m=1.03$ and solid (red) curves in
the middle and right panels denote $V$ with $b=1.01b_{\mathrm{m}}$ and
$b=0.99b_{\mathrm{m}}$, respectively.

In the strong deflection limit $b\rightarrow b_{\mathrm{m}}+0$, the exact
analytical expressions of $\bar{a}$ and $\bar{b}$ in the deflection angle (1)
has been obtained as

$\bar{a}=\frac{r_{\mathrm{m}}}{\sqrt{3mr_{\mathrm{m}}-4q^{2}}}$ (16)

and

$\displaystyle\bar{b}$ $\displaystyle=$
$\displaystyle\bar{a}\log\left[\frac{8(3mr_{\mathrm{m}}-4q^{2})^{3}}{m^{2}r_{\mathrm{m}}^{2}(mr_{\mathrm{m}}-q^{2})^{2}}\right.$
$\displaystyle\left.\times\left(2\sqrt{mr_{\mathrm{m}}-q^{2}}-\sqrt{3mr_{\mathrm{m}}-4q^{2}}\right)^{2}\right]-\pi,$

respectively, by Tsukamoto Tsukamoto:2016jzh ; Tsukamoto:2016oca . They
recovers the results by Eiroa et al. Eiroa:2002mk  and Bozza Bozza:2002zj  as
shown Refs Tsukamoto:2016jzh ; Tsukamoto:2016oca . Figure 3 shows the
parameters $\bar{a}$ and $\bar{b}$.

Figure 3: Parameters $\bar{a}$, $\bar{b}$, $\bar{c}$, and $\bar{d}$ in Eqs.
(1) and (2) in the strong deflection limits. Solid (red), dashed (green),
dotted (magenta), and long-dashed (cyan) curves denote $\bar{a}$, $\bar{b}$,
$\bar{c}$, and $\bar{d}$, respectively.

###  II.2 Light rays slightly inside of the photon sphere

We consider that light rays are reflected by a potential barrier near the
antiphoton sphere for $m<q<3m/(2\sqrt{2})$. The example of the effective
potential is plotted as the right panel in Fig. 2. The exact forms of
$\bar{c}$ and $\bar{d}$ of the deflection angle (2) in the strong deflection
limit $b\rightarrow b_{\mathrm{m}}-0$ is given by Tsukamoto:2021fsz

$\bar{c}\equiv\frac{2r_{\mathrm{m}}}{\sqrt{3mr_{\mathrm{m}}-4q^{2}}}$ (18)

and

$\displaystyle\bar{d}=$
$\displaystyle\bar{c}\log\left[\frac{2\left(3mr_{\mathrm{m}}-4q^{2}\right)}{r_{\mathrm{c}}(mr_{\mathrm{m}}-q^{2})}\left(\frac{r_{\mathrm{m}}}{r_{\mathrm{c}}}-1\right)\right.$
$\displaystyle\left.\times\frac{-4q^{2}(r_{\mathrm{m}}+r_{\mathrm{c}})+2mr_{\mathrm{m}}(r_{\mathrm{m}}+2r_{c})}{4(mr_{\mathrm{m}}-q^{2})+\sqrt{2(3mr_{\mathrm{m}}-4q^{2})(mr_{\mathrm{m}}-q^{2})}}\right]-\pi.$

The parameters $\bar{c}$ and $\bar{d}$ are plotted in Figure 3.

##  III Retrolensing in the Reissner-Nordström spacetime

In this section, investigated retrolensing in the Reissner-Nordström spacetime
as follow Ref. Bozza:2004kq . As shown in Fig. 4, we consider a lens
configuration that the ray is emitted by the sun S, it is reflected by the
photon sphere as a lens L with a deflection angle $\alpha$ and it reaches an
observer O. An image I with an image angle $\theta$ can be seen by the
observer O. We define effective deflection angle $\bar{\alpha}$ as

$\bar{\alpha}\equiv\alpha-2\pi n,$ (20)

where $n$ is a winding number of the ray.

Figure 4: Lens Configuration. The ray of the sun S at the source angle
$\beta\equiv\angle$OLS is reflected by a photon sphere L with an effective
deflection angle $\bar{\alpha}$. $\theta$ denotes image angle of image I
observed by an observer O and $\bar{\theta}$ denotes an angle between the line
LS and the ray at S.

By using a source angle $\beta\equiv\angle$OLS defined in the domain 0
$\leq\beta\leq\pi$ and an angle $\bar{\theta}$ between the line LS and the
light ray at S, the Ohanian lens equation Ohanian_1987 ; Bozza:2004kq ;
Bozza:2008ev  is expressed by

$\beta=\pi-\bar{\alpha}(\theta)+\theta+\bar{\theta}.$ (21)

We assume that the photon sphere L, the observer O, and the sun S are almost
aligned in this order. This yields

$\beta\sim 0,$ (22) $\bar{\alpha}\sim\pi,$ (23) $\alpha\sim\pi+2\pi n,$ (24)

and

$D_{\mathrm{ls}}=D_{\mathrm{ol}}+D_{\mathrm{os}},$ (25)

where $D_{\mathrm{ls}}$, $D_{\mathrm{ol}}$, and $D_{\mathrm{os}}$ are the
distances between L and S, between O and L, and between O and S, respectively.
We assume $b_{\mathrm{m}}\ll D_{\mathrm{ol}}$ and $b_{\mathrm{m}}\ll
D_{\mathrm{ls}}$ and we neglect $\theta=b/D_{\mathrm{ol}}$ and
$\bar{\theta}=b/D_{\mathrm{ls}}$ in the lens equation.

###  III.1 Light rays slightly outside of the photon sphere

From the deflection angle $\alpha(b)$ (1) and $b=\theta D_{\mathrm{ol}}$, we
obtain the positive solution of the lens equation (21) as
$\theta=\theta_{n\mathrm{out}}(\beta)$, where

$\theta_{n\mathrm{out}}(\beta)\equiv\theta_{\mathrm{m}}\left\\{1+\exp\left[\frac{\bar{b}-(1+2n)\pi+\beta}{\bar{a}}\right]\right\\},$
(26)

and where $\theta_{\mathrm{m}}\equiv b_{\mathrm{m}}/D_{\mathrm{ol}}$ is the
image angle of the photon sphere. The magnification $\mu_{n\mathrm{out}}$ of
the image is given by

$\mu_{n\mathrm{out}}(\beta)=-\frac{D_{\mathrm{os}}^{2}}{D_{\mathrm{ls}}^{2}}s(\beta)\theta_{n\mathrm{out}}\frac{d\theta_{n\mathrm{out}}}{d\beta},$
(27)

where for a point source $s(\beta)$ is given by

$s(\beta)=\frac{1}{\beta}$ (28)

and for an uniform-luminous disk with a finite size on the observer’s sky
Witt:1994 ; Nemiroff:1994uz ; Alcock:1997fi , it is given by, an integral over
the disk on a source plane,

$s(\beta)=\frac{1}{\pi\beta_{\mathrm{s}}^{2}}\int_{\mathrm{disk}}d\beta^{\prime}d\phi,$
(29)

where $\beta^{\prime}$ is a reduced radial coordinate which is divided by
$D_{\mathrm{ls}}$ on the source plane, $\beta_{\mathrm{s}}\equiv
R_{\mathrm{s}}/D_{\mathrm{ls}}$ is the reduced radius of the sun,
$R_{\mathrm{s}}$ is the radius of the sun, and where $\phi$ is an azimuthal
coordinate around the origin on the source plane.

By fixing the intersection point of an axis $\beta=0$ and the source plane as
the origin of the coordinates on the source plane, $s(\beta)$ can be rewritten
as

$\displaystyle s(\beta)$ $\displaystyle=$
$\displaystyle\frac{2}{\pi\beta_{\mathrm{s}}^{2}}\left[\pi(\beta_{\mathrm{s}}-\beta)\right.$
(30)
$\displaystyle\left.+\int^{\beta+\beta_{\mathrm{s}}}_{-\beta+\beta_{\mathrm{s}}}\arccos\frac{\beta^{2}+\beta^{\prime
2}-\beta^{2}_{\mathrm{s}}}{2\beta\beta^{\prime}}d\beta^{\prime}\right]$

for $\beta\leq\beta_{\mathrm{s}}$ and

$\displaystyle
s(\beta)=\frac{2}{\pi\beta_{\mathrm{s}}^{2}}\int^{\beta+\beta_{\mathrm{s}}}_{\beta-\beta_{\mathrm{s}}}\arccos\frac{\beta^{2}+\beta^{\prime
2}-\beta^{2}_{\mathrm{s}}}{2\beta\beta^{\prime}}d\beta^{\prime}$ (31)

for $\beta_{\mathrm{s}}\leq\beta$. If the photon sphere, the observer, and the
sun are perfectly aligned, $s(\beta)$ is given by

$s(0)=\frac{2}{\beta_{\mathrm{s}}}.$ (32)

From Eqs. (26) and (27), its magnification $\mu_{n\mathrm{out}}(\beta)$ is
obtained as

$\displaystyle\mu_{n\mathrm{out}}(\beta)$ $\displaystyle=$
$\displaystyle-\frac{D_{\mathrm{os}}^{2}}{D_{\mathrm{ls}}^{2}}\frac{\theta_{\mathrm{m}}^{2}e^{\left[\bar{b}-(1+2n)\pi\right]/\bar{a}}}{\bar{a}}$
(33)
$\displaystyle\times\left\\{1+e^{\left[\bar{b}-(1+2n)\pi\right]/\bar{a}}\right\\}s(\beta).$

Notice that the lens equation has a negative solution
$\theta\sim-\theta_{n\mathrm{out}}(\beta)$ and its magnification is given by
$-\mu_{n\mathrm{out}}(\beta)$ approximately. The total magnification
$\mu_{\mathrm{totout}}(\beta)$ of the couples of the images from $n=0$ to
$\infty$ is given by

$\displaystyle\mu_{\mathrm{totout}}(\beta)$ $\displaystyle\equiv$
$\displaystyle 2\sum_{n=0}^{\infty}\left|\mu_{n\mathrm{out}}(\beta)\right|$
$\displaystyle=$ $\displaystyle
2\frac{D_{\mathrm{os}}^{2}}{D_{\mathrm{ls}}^{2}}\frac{\theta_{\mathrm{m}}^{2}}{\bar{a}}\left|s(\beta)\right|\left[\frac{e^{(\bar{b}-\pi)/\bar{a}}}{1-e^{-2\pi/\bar{a}}}+\frac{e^{2(\bar{b}-\pi)/\bar{a}}}{1-e^{-4\pi/\bar{a}}}\right]$

and it gives, in a perfect-aligned case,

$\mu_{\mathrm{totout}}(0)=4\frac{D_{\mathrm{os}}^{2}}{D_{\mathrm{ls}}^{2}}\frac{\theta_{\mathrm{m}}^{2}}{\bar{a}\beta_{\mathrm{s}}}\left[\frac{e^{(\bar{b}-\pi)/\bar{a}}}{1-e^{-2\pi/\bar{a}}}+\frac{e^{2(\bar{b}-\pi)/\bar{a}}}{1-e^{-4\pi/\bar{a}}}\right].$
(35)

We consider retrolensing of the sun light reflected by the photon sphere at
$D_{\mathrm{ol}}=0.01$pc away as shown in Fig. 5. We assume that the sun moves
with the orbital velocity $v=30$km/s on a source plane and with the closest
separation $\beta_{\mathrm{min}}$ between the center of the sun disk and the
axis $\beta=0$ on the source plane as shown Fig. 5.

Figure 5: Retrolensing. Let the source plane a plane that the sun is on and
that is orthogonal to the optical axis $\beta=0$. $\beta_{\mathrm{min}}$
denotes the smallest source angle. We assume the sun moves with the orbital
velocity $v=30$km/s on the source plane.

The separation between the image $\theta_{n\mathrm{out}}$ and its paired image
is given by

$2\theta_{n\mathrm{out}}=2\theta_{\mathrm{m}}\left\\{1+\exp\left[\frac{\bar{b}-(1+2n)\pi+\beta}{\bar{a}}\right]\right\\}$
(36)

and the separation of the image angles with the winding number $n=0$ as a
function of $q/m$ is plotted in Fig. 6.

Figure 6: The image separation $2\theta_{0\mathrm{out}}$ of outer images as a
function of $q/m$. Solid (red), dashed (green), and dotted (magenta) curves
denote the image separation in the cases of $m=60M_{\odot}$, $30M_{\odot}$,
and $10M_{\odot}$, respectively. The distance to the photon sphere is
$D_{\mathrm{ol}}=0.01$pc and the source angle is $\beta=0$ and the winding
number is $n=0$.

###  III.2 Light rays slightly inside of the photon sphere

For the deflection angle (2), the positive solution of the lens equation (21)
is obtained by $\theta=\theta_{n\mathrm{in}}(\beta)$, where
$\theta_{n\mathrm{in}}(\beta)$ is given by

$\theta_{n\mathrm{in}}(\beta)\equiv\frac{\theta_{\mathrm{m}}}{1+\exp\left[\frac{\bar{d}-(1+2n)\pi+\beta}{\bar{c}}\right]}$
(37)

and its magnification is given by

$\mu_{n\mathrm{in}}(\beta)=-\frac{D_{\mathrm{os}}^{2}}{D_{\mathrm{ls}}^{2}}\frac{\theta_{\mathrm{m}}^{2}\exp\left[\frac{\bar{d}-(1+2n)\pi}{\bar{c}}\right]}{\bar{c}\left\\{1+\exp\left[\frac{\bar{d}-(1+2n)\pi}{\bar{c}}\right]\right\\}^{3}}s(\beta).$
(38)

A negative solution of the lens equation is
$\theta=\theta_{-n\mathrm{in}}(\beta)$, where

$\theta_{-n\mathrm{in}}(\beta)=-\theta_{n\mathrm{in}}(-\beta)\sim-\theta_{n\mathrm{in}}(\beta)$
(39)

and its magnification is given by $-\mu_{n\mathrm{in}}(\beta)$. The total
magnification $\mu_{\mathrm{totin}}(\beta)$ of the pair images from $n=0$ to
$\infty$ is given by

$\displaystyle\mu_{\mathrm{totin}}(\beta)$ $\displaystyle=$ $\displaystyle
2\sum_{n=0}^{\infty}\left|\mu_{n\mathrm{in}}(\beta)\right|$ $\displaystyle=$
$\displaystyle
2\sum_{n=0}^{\infty}\frac{D_{\mathrm{os}}^{2}}{D_{\mathrm{ls}}^{2}}\frac{\theta_{\mathrm{m}}^{2}e^{\left[\bar{d}-(1+2n)\pi\right]/\bar{c}}}{\bar{c}\left\\{1+e^{\left[\bar{d}-(1+2n)\pi\right]/\bar{c}}\right\\}^{3}}\left|s(\beta)\right|.$

and it gives, in a perfect-aligned case,

$\mu_{\mathrm{totin}}(0)=4\sum_{n=0}^{\infty}\frac{D_{\mathrm{os}}^{2}}{D_{\mathrm{ls}}^{2}}\frac{\theta_{\mathrm{m}}^{2}e^{\left[\bar{d}-(1+2n)\pi\right]/\bar{c}}}{\bar{c}\left\\{1+e^{\left[\bar{d}-(1+2n)\pi\right]/\bar{c}}\right\\}^{3}\beta_{\mathrm{s}}}.$
(41)

The image separation between $\theta_{n\mathrm{in}}$ and its pair is given by

$2\theta_{n\mathrm{in}}=\frac{2\theta_{\mathrm{m}}}{1+\exp\left[\frac{\bar{d}-(1+2n)\pi+\beta}{\bar{c}}\right]}$
(42)

and the case of $n=0$ is shown in Fig. 7.

Figure 7: The image separation $2\theta_{0\mathrm{in}}$ of the inner images as
a function of $q/m$. Solid (red), dashed (green), and dotted (magenta) curves
denote the image separation in the cases of $m=60M_{\odot}$, $30M_{\odot}$,
and $10M_{\odot}$, respectively. The distance to the photon sphere is
$D_{\mathrm{ol}}=0.01$pc and the source angle is $\beta=0$ and the winding
number is $n=0$.

###  III.3 Retrolensing curves

The light curves of retrolensing by the photon sphere in the Reissner-
Nordström black hole and naked singularity spacetimes are shown in Figs. 8-10.
Figure 11 shows the the apparent magnitude of the light curves at the peak in
the perfect-aligned case as a function of $q/m$. Notice that light curves
consist of only light rays reflected slightly outside of the photon sphere
around the black hole for $q/m\leq 1$ while the ones consist of both light
rays reflected slightly outside and inside of the photon sphere around the
naked singularity for $1<q/m<3/(2\sqrt{2})$.

Figure 8: Light curves with the charge $q/m=0$, $1$, $1.01$, and $1.04$ are
denoted by (red) solid, (green) dashed, (magenta) dotted, and (blue) dash-
dotted curves, respectively. We set the distance to the lens
$D_{\mathrm{ol}}=0.01$pc, the mass $m=30M_{\odot}$, and the smallest source
angle $\beta_{\mathrm{min}}=0$. Figure 9: Light curves with the smallest
source angle $\beta_{\mathrm{min}}=0$, $0.5\beta_{\mathrm{s}}$,
$\beta_{\mathrm{s}}$, and $1.5\beta_{\mathrm{s}}$ are denoted by (red) solid,
(green) dashed, (magenta) dotted, and (blue) dash-dotted curves, respectively.
We set the distance to the lens $D_{\mathrm{ol}}=0.01$pc, the mass
$m=30M_{\odot}$, and the charge $q/m=1.01$. Figure 10: Light curves with the
mass $m=10M_{\odot}$, $30M_{\odot}$, and $60M_{\odot}$ are denoted by (red)
solid, (green) dashed, and (magenta) dotted curves, respectively. We set the
distance to the lens $D_{\mathrm{ol}}=0.01$pc, and the charge $q/m=1.01$, and
the smallest source angle $\beta_{\mathrm{min}}=0$. Figure 11: The apparent
magnitudes of the peak in the perfect-aligned ($\beta_{\mathrm{min}}=0$) of a
photon sphere at a distance $D_{\mathrm{ol}}=0.01$pc as a function of $q/m$.
The apparent magnitudes of the peak for the mass with $m=60M_{\odot}$,
$30M_{\odot}$, and $10M_{\odot}$ are shown as (red) solid, (green) dashed, and
(magenta) dotted curves, respectively.

##  IV Discussion and Conclusion

We have investigated retrolensing of the sun light reflected by a photon
sphere in a Reissner-Nordström black hole and naked singularity spacetimes. We
have shown that the apparent magnitude of the retrolensing depends strongly on
the specific charge $q/m$ in the case of naked singularity for
$1<q/m<3/(2\sqrt{2})$. It is an interesting feature of retrolensing in the
Reissner-Nordström naked singularity spacetime because the apparent magnitude
of the retrolensing in the Reissner-Nordström black hole spacetime for $q/m<1$
almost does not depend on the specific charge $q/m$ as shown by Tsukamoto and
Gong Tsukamoto:2016oca . The retrolensing by the photon sphere around the
Reissner-Nordström naked singularity can be brighter than the one around a
Reissner-Nordström black hole because the light rays which reflected by the
potential barrier near an antiphoton sphere also reach into an observer.

The light curve of the retrolensing has a characteristic shape as shown in
Figs. 8-10 because of the spherical symmetry and the disk shape of the sun.
The time-symmetrical shape of the light curves helps us to distinguish the
retrolensing from other phenomena of light emissions. And we would distinguish
the event of retrolensing from the other event since the retrolensing light
curve has precise solar spectra and it can be observed on the ecliptic
Holz:2002uf .

We do not consider the case of $q/m=3/(2\sqrt{2})$ on this paper. For
$q/m=3/(2\sqrt{2})$, the effective potential gives
$V(r_{\mathrm{m}})=V^{\prime}(r_{\mathrm{m}})=V^{\prime\prime}(r_{\mathrm{m}})=0$
and $V^{\prime\prime\prime}(r_{\mathrm{m}})<0$ and the photon sphere and
antiphoton photon sphere come together as one marginally unstable photon
sphere at $r=r_{\mathrm{m}}=r_{\mathrm{aps}}=3m/2$. In this case, the
deflection angle diverges nonlogarithmically and we cannot apply the formulas
obtained in this paper to the marginally unstable photon sphere. The
deflection angle in the strong deflection limit $b\rightarrow
b_{\mathrm{m}}+0$ of the marginally unstable photon sphere of the Reissner-
Nordström spacetime and its application to a usual lens configuration have
been investigated by Tsukamoto Tsukamoto:2020iez . The deflection angle of the
marginally unstable photon sphere in the strong deflection limit $b\rightarrow
b_{\mathrm{m}}-0$ and retrolensing by the marginal unstable photon sphere in
the strong deflection limits $b\rightarrow b_{\mathrm{m}}-0$ and $b\rightarrow
b_{\mathrm{m}}+0$ are left as future work.

## References

  * (1) P. Schneider, J. Ehlers, and E. E. Falco, Gravitational Lenses (Springer-Verlag, Berlin, 1992).
  * (2) P. Schneider, C. S. Kochanek, and J. Wambsganss, Gravitational Lensing: Strong, Weak and Micro, Lecture Notes of the 33rd Saas-Fee Advanced Course, edited by G. Meylan, P. Jetzer, and P. North (Springer-Verlag, Berlin, 2006).
  * (3) Y. Hagihara, Jpn. J. Astron. Geophys., 8, 67 (1931).
  * (4) C. Darwin, Proc. R. Soc. Lond. A 249, 180 (1959).
  * (5) R. d’ E. Atkinson, Astron. J. 70, 517 (1965).
  * (6) J.-P. Luminet, Astron. Astrophys. 75, 228 (1979).
  * (7) H. C. Ohanian, Am. J. Phys. 55, 428 (1987).
  * (8) R. J. Nemiroff, Am. J. Phys. 61, 619 (1993).
  * (9) S. Frittelli, T. P. Kling, and E. T. Newman, Phys. Rev. D 61, 064021 (2000).
  * (10) K. S. Virbhadra and G. F. R. Ellis, Phys. Rev. D 62, 084003 (2000).
  * (11) V. Bozza, S. Capozziello, G. Iovane, and G. Scarpetta, Gen. Relativ. Gravit. 33, 1535 (2001).
  * (12) V. Bozza, Phys. Rev. D 66, 103001 (2002).
  * (13) V. Perlick, Phys. Rev. D 69, 064017 (2004).
  * (14) K. S. Virbhadra, Phys. Rev. D 79, 083004 (2009).
  * (15) V. Bozza, Gen. Relativ. Gravit. 42, 2269 (2010).
  * (16) N. Tsukamoto and T. Harada, Phys. Rev. D 95, 024030 (2017).
  * (17) R. Shaikh, P. Banerjee, S. Paul, and T. Sarkar, JCAP 1907, 028 (2019).
  * (18) R. Shaikh, P. Banerjee, S. Paul, and T. Sarkar, Phys. Rev. D 99, 104040 (2019).
  * (19) N. Tsukamoto, Phys. Rev. D 101, 104021 (2020).
  * (20) N. Tsukamoto, Phys. Rev. D 102, 104029 (2020).
  * (21) S. Paul, Phys. Rev. D 102, 064045 (2020).
  * (22) B. P. Abbott et al. [LIGO Scientific and Virgo Collaborations], Phys. Rev. Lett. 116, 061102 (2016).
  * (23) K. Akiyama et al. [Event Horizon Telescope Collaboration], Astrophys. J. 875, L1 (2019).
  * (24) V. Perlick, Living Rev. Relativity 7, 9 (2004).
  * (25) C. M. Claudel, K. S. Virbhadra, and G. F. R. Ellis, J. Math. Phys. 42, 818 (2001).
  * (26) V. Perlick and O. Y. Tsupko, [arXiv:2105.07101 [gr-qc]].
  * (27) S. Hod, Phys. Lett. B 727, 345 (2013); S. Hod, Phys. Lett. B 776, 1 (2018).
  * (28) N. G. Sanchez, Phys. Rev. D 18, 1030 (1978); Y. Decanini, A. Folacci, and B. Raffaelli, Phys. Rev. D 81, 104039 (2010); S. W. Wei, Y. X. Liu, and H. Guo, Phys. Rev. D 84, 041501 (2011).
  * (29) W. H. Press, Astrophys. J. 170, L105 (1971); C. J. Goebel, Astrophys. J. 172, L95 (1972); I. Z. Stefanov, S. S. Yazadjiev, and G. G. Gyulchev, Phys. Rev. Lett. 104, 251103 (2010); B. Raffaelli, Gen. Rel. Grav. 48, 16 (2016).
  * (30) M. A. Abramowicz and A. R. Prasanna, Mon. Not. Roy. Astr. Soc. 245, 720 (1990); M. A. Abramowicz, Mon. Not. Roy. Astr. Soc. 245, 733 (1990); B. Allen, Nature 347, 615 (1990); W. Hasse and V. Perlick, Gen. Relativ. Gravit. 34, 415 (2002).
  * (31) P. Mach, E. Malec, and J. Karkowski, Phys. Rev. D 88, 084056 (2013); E. Chaverra and O. Sarbach, Class. Quant. Grav. 32, 155006 (2015); M. Cvetic, G. W. Gibbons, and C. N. Pope, Phys. Rev. D 94, 106005 (2016); Y. Koga and T. Harada, Phys. Rev. D 94, 044053 (2016); Y. Koga and T. Harada, Phys. Rev. D 98, 024018 (2018); Y. Koga, Phys. Rev. D 99, 064034 (2019).
  * (32) C. Barcelo and M. Visser, Nucl. Phys. B 584, 415 (2000); Y. Koga, Phys. Rev. D 101 104022 (2020).
  * (33) W. L. Ames and K. S. Thorne, Astrophys. J. 151, 659 (1968); J. L. Synge, Mon. Not. Roy. Astron. Soc. 131, 463 (1966); H. Yoshino, K. Takahashi, and K. i. Nakao, Phys. Rev. D 100, 084062 (2019).
  * (34) G. W. Gibbons and C. M. Warnick, Phys. Lett. B 763, 169 (2016); P. V. P. Cunha, C. A. R. Herdeiro, and E. Radu, Phys. Rev. D 96, 024039 (2017); T. Shiromizu, Y. Tomikawa, K. Izumi, and H. Yoshino, PTEP 2017, 033E01 (2017); H. Yoshino, K. Izumi, T. Shiromizu, and Y. Tomikawa, PTEP 2017, 063E01 (2017); D. V. Gal’tsov and K. V. Kobialko, Phys. Rev. D 99, 084043 (2019). D. V. Gal’tsov and K. V. Kobialko, Phys. Rev. D 100, 104005 (2019); Y. Koga and T. Harada, Phys. Rev. D 100, 064040 (2019); M. Siino, Class. Quantum Grav. 38, 025005 (2021); H. Yoshino, K. Izumi, T. Shiromizu, and Y. Tomikawa, PTEP 2020, 023E02 (2020); L. M. Cao and Y. Song, [arXiv:1910.13758 [gr-qc]]; H. Yoshino, K. Izumi, T. Shiromizu, and Y. Tomikawa, PTEP 2020, 053E01 (2020); K. Lee, T. Shiromizu, H. Yoshino, K. Izumi, and Y. Tomikawa, PTEP 2020, 103E03 (2020); K. Izumi, Y. Tomikawa, T. Shiromizu, and H. Yoshino, PTEP, 2021, 083E02 (2021); M. Siino, [arXiv:2107.06551 [gr-qc]].
  * (35) J. Keir, Class. Quant. Grav. 33, 135009 (2016).
  * (36) V. Cardoso, L. C. B. Crispino, C. F. B. Macedo, H. Okawa, and P. Pani, Phys. Rev. D 90, 044069 (2014).
  * (37) P. V. P. Cunha, E. Berti, and C. A. R. Herdeiro, Phys. Rev. Lett. 119, 251102 (2017).
  * (38) V. Bozza, Phys. Rev. D 67, 103006 (2003).
  * (39) E. F. Eiroa, G. E. Romero, and D. F. Torres, Phys. Rev. D 66, 024010 (2002).
  * (40) A. O. Petters, Mon. Not. Roy. Astron. Soc. 338, 457 (2003).
  * (41) E. F. Eiroa and D. F. Torres, Phys. Rev. D 69, 063004 (2004).
  * (42) V. Bozza and L. Mancini, Astrophys. J. 611, 1045 (2004).
  * (43) V. Bozza, F. De Luca, G. Scarpetta, and M. Sereno, Phys. Rev. D 72, 083003 (2005).
  * (44) V. Bozza and M. Sereno, Phys. Rev. D 73, 103004 (2006).
  * (45) V. Bozza, F. De Luca, and G. Scarpetta, Phys. Rev. D 74, 063001 (2006).
  * (46) S. V. Iyer and A. O. Petters, Gen. Rel. Grav. 39, 1563 (2007).
  * (47) V. Bozza and G. Scarpetta, Phys. Rev. D 76, 083008 (2007).
  * (48) N. Tsukamoto, Phys. Rev. D 94, 124001 (2016).
  * (49) A. Ishihara, Y. Suzuki, T. Ono, and H. Asada, Phys. Rev. D 95, 044017 (2017).
  * (50) N. Tsukamoto and Y. Gong, Phys. Rev. D 95, 064034 (2017).
  * (51) N. Tsukamoto, Phys. Rev. D 95, 064035 (2017).
  * (52) N. Tsukamoto, Phys. Rev. D 95, 084021 (2017).
  * (53) G. F. Aldi and V. Bozza, JCAP 02, 033 (2017).
  * (54) T. Hsieh, D. S. Lee, and C. Y. Lin, Phys. Rev. D 103, 104063 (2021).
  * (55) K. Takizawa and H. Asada, Phys. Rev. D 103, 104039 (2021).
  * (56) N. Tsukamoto, Phys. Rev. D 103, 024033 (2021).
  * (57) N. Tsukamoto, [arXiv:2105.14336 [gr-qc]].
  * (58) F. Aratore and V. Bozza, [arXiv:2107.05723 [gr-qc]].
  * (59) T. Chiba and M. Kimura, PTEP 2017, 043E01 (2017).
  * (60) A. de Vries, Class. Quantum Grav. 17, 123 (2000).
  * (61) R. Takahashi, Publ. Astron. Soc. Jap. 57, 273 (2005).
  * (62) A. F. Zakharov, Phys. Rev. D 90, 062007 (2014).
  * (63) K. Akiyama et al. [Event Horizon Telescope], Astrophys. J. Lett. 875, L6 (2019).
  * (64) P. Kocherlakota et al. [Event Horizon Telescope], Phys. Rev. D 103, 104047 (2021).
  * (65) M. Sereno, Phys. Rev. D 69, 023002 (2004).
  * (66) A. Y. Bin-Nun, Phys. Rev. D 82, 064009 (2010).
  * (67) A. Y. Bin-Nun, Class. Quant. Grav. 28, 114003 (2011).
  * (68) N. Tsukamoto, [arXiv:2107.07146 [gr-qc]].
  * (69) D. E. Holz and J. A. Wheeler, Astrophys. J. 578, 330 (2002).
  * (70) F. De Paolis, G. Ingrosso, A. Geralico, and A. A. Nucita, Astron. Astrophys. 409, 809 (2003).
  * (71) F. De Paolis, A. Geralico, G. Ingrosso, A. A. Nucita, and A. Qadir, Astron. Astrophys. 415, 1 (2004).
  * (72) A. Abdujabbarov, B. Ahmedov, N. Dadhich and F. Atamurotov, Phys. Rev. D 96, 084017 (2017).
  * (73) G. Zaman Babar, F. Atamurotov, and A. Zaman Babar, [arXiv:2104.01340 [gr-qc]].
  * (74) V. Bozza, Phys. Rev. D 78, 103005 (2008).
  * (75) H. J. Witt and S. Mao, ApJ, 430, 505 (1994).
  * (76) R. J. Nemiroff and W. A. D. T. Wickramasinghe, Astrophys. J. 424, L21 (1994).
  * (77) C. Alcock et al. [MACHO and GMAN Collaborations], Astrophys. J. 491, 436 (1997).