# Canonical theorems in geometric Ramsey theory

Panna Gehér Eötvös Loránd University, Budapest, Hungary. Email:
<EMAIL_ADDRESS>Arsenii Sagdeev Alfréd Rényi Institute of
Mathematics, Budapest, Hungary. Email<EMAIL_ADDRESS>Géza Tóth
Alfréd Rényi Institute of Mathematics, Budapest, Hungary. Email:
<EMAIL_ADDRESS>
###### Abstract

In Euclidean Ramsey Theory usually we are looking for monochromatic
configurations in the Euclidean space, whose points are colored with a fixed
number of colors. In the canonical version, the number of colors is arbitrary,
and we are looking for an ‘unavoidable’ set of colorings of a finite
configuration, that is a set of colorings with the property that one of them
always appears in any coloring of the space. This set definitely includes the
monochromatic and the rainbow colorings. In the present paper, we prove the
following two results of this type. First, for any acute triangle $T$, and any
coloring of $\mathbb{R}^{3}$, there is either a monochromatic or a rainbow
copy of $T$. Second, for every $m$, there exists a sufficiently large $n$ such
that in any coloring of $\mathbb{R}^{n}$, there exists either a monochromatic
or a rainbow $m$-dimensional unit hypercube. In the maximum norm,
$\ell_{\infty}$, we have a much stronger statement. For every finite $M$,
there exits an $n$ such that in any coloring of $\mathbb{R}_{\infty}^{n}$,
there is either a monochromatic or a rainbow isometric copy of $M$.

Key words: Euclidean Ramsey theory, canonical Ramsey theorem, coloring of the
space

Mathematics Subject Classification: 05D10, 05C55

##  1 Introduction

A typical problem in Ramsey theory [23] is to show that every $r$-coloring of
a sufficiently large ‘domain’ contains a monochromatic copy of some fixed
‘configuration’. A classical result in the field is the van der Waerden
theorem [42], which states that for all $k,r\in\mathbb{N}$, there exists $n$
such that every $r$-coloring of $[n]$ contains a monochromatic $k$-term
arithmetic progression. Each problem in Ramsey theory has its canonical
counterpart, in which the goal is to list a minimal set $S$ of colorings which
is ‘unavoidable’ regardless of the number of colors in use. That is, in any
coloring of the domain we have a configuration whose coloring belongs to $S$.
A coloring of a configuration is called rainbow if all of its point are of
different color. For example, the canonical van der Waerden [8] theorem states
that for all $k\in\mathbb{N}$, there exists $n$ such that every coloring of
$[n]$ contains either a monochromatic or a rainbow $k$-term arithmetic
progression. For modern exposition and further generalizations of this result,
see [20, 21] and the references therein. We note that the minimal sets of
unavoidable colorings often contain other colorings along with the
monochromatic and the rainbow. In fact, this is the case for canonical
versions of both the Ramsey theorem due to Erdős and Rado [12], see also [26,
1, 43, 25], and the Hales–Jewett theorem [37, 35], see Section 3.1 for the
exact statement.

In their celebrated trilogy [9, 10, 11], Erdős, Graham, Montgomery,
Rothschild, Spencer and Straus initiated a systematic study of the following
general problem in geometric Ramsey theory. For $n,r\in\mathbb{N}$ and
$A\subset\mathbb{R}^{n}$, does every $r$-coloring of $\mathbb{R}^{n}$ contain
a monochromatic isometric copy of $A$? They also introduced the notation
$\mathbb{R}^{n}\xrightarrow{r}A$ for an affirmative answer on the latter
question. Note that only a few results in the field are tight. Even the
innocent-looking famous special case, when $n=2$ and $A=I$ is a fixed two-
point set is open. It is equivalent to the notorious Hadwiger-Nelson problem
which asks for the chromatic number of the plane. After 75 years of extensive
study, there are still two values of $r$, namely $r=5$ and $r=6$, for which we
do not know whether $\mathbb{R}^{2}\xrightarrow{r}I$ or not, see [7, 14] and
the survey in [40].

Another major open problem in the field is to describe all Ramsey sets, that
is sets $A$ with the property that for all $r\in\mathbb{N}$ there is a
sufficiently large $n$ with $\mathbb{R}^{n}\xrightarrow{r}A$. It is known that
every Ramsey set is spherical [9, Theorem 13] and finite [10, Theorem 19]. We
refer the reader to [32] for the discussion about if these two necessary
conditions might also be sufficient. From the other direction, it is known
[27, 2] that (the vertex sets of) all regular polytopes are Ramsey. The vertex
set $A$ of a box or a simplex in arbitrary dimension has an even stronger
exponentially Ramsey property: there exists a positive constant $c(A)$ such
that for all $r\in\mathbb{N}$ and $n\sim c(A)\log r$, we have
$\mathbb{R}^{n}\xrightarrow{r}A$, see [17]. For the currently best bounds on
$c(A)$, we refer the reader to [31] and the references therein.

To the best of our knowledge, the study of canonical results in Euclidean
Ramsey theory was initiated only recently. In [33] the authors proved that
every $r$-coloring of $\mathbb{R}^{n}$ contains either a monochromatic or a
rainbow copy of a fixed rectangle for $n=\Theta(r)$. However, this result is
far from optimal, because $n=\Theta(\log r)$ dimensions are already sufficient
to ensure a monochromatic copy of a given rectangle as we mentioned above. In
fact, it seems hard to avoid rainbow copies of a given set $A$ whenever the
number of colors in use is much larger than the cardinality $|A|$. Thus it
feels only natural to expect that for most of the Ramsey sets $A$, if not for
all of them, there exists a finite $n$ such that every coloring of
$\mathbb{R}^{n}$ contains either a monochromatic or a rainbow copy of $A$
regardless of the number of colors in use. In the spirit of classic arrow
notation, we denote the latter claim by
$\mathbb{R}^{n}\smash{\xrightarrow[\smash{\raisebox{3.01385pt}{\tiny$\mathbf{MR}$}}]{}}A$
for short. Recently Cheng and Xu [4] found the first evidence supporting this
intuition. Using our notation, their theorem can be stated as follows:

######  Theorem 1 (Cheng, Xu [4]).

a

  * •

For every acute triangle $T$, there exists $n$ such that
$\mathbb{R}^{n}\smash{\xrightarrow[\smash{\raisebox{3.01385pt}{\tiny$\mathbf{MR}$}}]{}}T$.

a

  * •

For every right triangle $T$, we have
$\mathbb{R}^{3}\smash{\xrightarrow[\smash{\raisebox{3.01385pt}{\tiny$\mathbf{MR}$}}]{}}T$.

In fact, they proved a stronger statement, which also covers simplices of
larger dimension and obtuse triangles that are not ‘too flat’. In the present
paper, we refine their technique and strengthen the above result.

######  Theorem 2.

For every acute triangle $T$, we have
$\mathbb{R}^{3}\smash{\xrightarrow[\smash{\raisebox{3.01385pt}{\tiny$\mathbf{MR}$}}]{}}T$.

Another canonical result obtained by Cheng and Xu in [4] covers the case of a
square $I^{2}$.

######  Theorem 3 (Cheng, Xu [4]).

There exists $n$ such that
$\mathbb{R}^{n}\smash{\xrightarrow[\smash{\raisebox{3.01385pt}{\tiny$\mathbf{MR}$}}]{}}I^{2}$.

Their method probably can be generalized to a cube $I^{3}$ but perhaps not
further, because for $m\geq 4$, diameter of the (vertex set of an)
$m$-dimensional hypercube $I^{m}$ becomes too large in comparison with its
side length, which seems to be an obstacle. Here we present a completely
different approach to bypass this limitation.

######  Theorem 4.

For every $m\in\mathbb{N}$, there exists $n$ such that
$\mathbb{R}^{n}\smash{\xrightarrow[\smash{\raisebox{3.01385pt}{\tiny$\mathbf{MR}$}}]{}}I^{m}$.

Euclidean Ramsey theory questions (along with the corresponding notation) can
be translated into non-Euclidean norms in a straightforward manner, with the
only difference being the definition of an isometric copy111For a norm $N$ on
$\mathbb{R}^{n}$, a set $A^{\prime}\subset\mathbb{R}^{n}$ is an $N$-isometric
copy of $A\subset\mathbb{R}^{n}$ if there exists a bijection $f:A\to
A^{\prime}$ such that
$\|\mathbf{x}-\mathbf{y}\|_{N}=\|f(\mathbf{x})-f(\mathbf{y})\|_{N}$ for all
$\mathbf{x},\mathbf{y}\in A$., see [13, 6, 15, 19, 28, 29, 39, 41]. The case
of an $n$-dimensional space $\mathbb{R}_{\infty}^{n}$ equipped with the max-
norm222Recall that for $\mathbf{x}=(x_{1},\dots,x_{n})\in\mathbb{R}^{n}$, its
max-norm is defined by $\|\mathbf{x}\|_{\infty}=\max_{i}|x_{i}|$ seems to
stand out here due to the following simple classification of finite
(exponentially) Ramsey sets. It is not hard to deduce from the Hales–Jewett
theorem that for every $r\in\mathbb{N}$ and a finite $M\subset\mathbb{R}^{d}$,
there exists $n$ such that $\mathbb{R}_{\infty}^{n}\xrightarrow{r}M$. In fact,
the latter claim holds already for $n=\Theta(\log r)$ as it was shown in [30,
16]. In the present paper, we deduce the following general canonical result in
max-norm Ramsey theory from the canonical Hales–Jewett theorem.

######  Theorem 5.

For every finite $M\subset\mathbb{R}^{d}$, there exists $n$ such that
$\mathbb{R}_{\infty}^{n}\smash{\xrightarrow[\smash{\raisebox{3.01385pt}{\tiny$\mathbf{MR}$}}]{}}M$.

Paper outline. In Section 2, we prove Theorems 2 and 4. In Section 3, we
formally state the canonical Hales–Jewett theorem and use it to prove 5.
Finally, in Section 4, we make some further comments and state more open
problems.

##  2 Euclidean norm

###  2.1 Triangles – proof of Theorem 2

Let $T$ be an acute triangle with sides $a$, $b$, $c$. Consider an arbitrary
coloring of $\mathbb{R}^{3}$ and assume for a contradiction that there is no
monochromatic or rainbow copy of $T$. Then some two points, which we denote by
$Y_{1}$ and $Y_{2}$, are of different colors. Take a sequence
$Y_{1}=X_{0},X_{1},\ldots,X_{k}=Y_{2}$ such that consecutive points are at
distance $c$. Then, there are two consecutive points, $X_{i}=A$ and
$X_{i+1}=B$, of different colors, say, $A$ is red and $B$ is blue.

We say that a pair of points $(X,Y)$ is a good pair if they are at distance
$c$, one of them is red while the other one is blue. In particular, $(A,B)$ is
a good pair. Let $X$, $Y$, $U$, $V$ be points such that $|XY|=|UV|=c$,
$|XU|=|YV|=b$, $|XV|=|YU|=a$. That is, $XYUV$ is a simplex, all of whose faces
are congruent to $T$. Then we say that the pair $(U,V)$ is an extension of the
pair $(X,Y)$.

######  Claim 1.

If $(X,Y)$ is a good pair, and $(U,V)$ is its extension, then $(U,V)$ is also
a good pair.

###### Proof.

Suppose without loss of generality that $X$ is red and $Y$ is blue. Since both
$UXY$ and $VXY$ are congruent to $T$, both $U$ and $V$ are red or blue. But
$UVX$ and $UVY$ are also congruent to $T$, so one of $U,V$ is red, and the
other one is blue. ∎

######  Claim 2.

Every good pair has an extension.

###### Proof.

Let $(A,B)$ be a good pair. Take a ball of radius $b$ (resp. $a$) about $A$
(resp. $B$). Their intersection is the circle $C_{1}$, its plane is
perpendicular to $AB$, and its center is on the segment $AB$ since $T$ is an
acute triangle. Similarly, take a ball of radius $b$ (resp. $a$) about $B$
(resp. $A$). Their intersection is the circle $C_{2}$, its plane is
perpendicular to $AB$, and its center is on the segment $AB$. Fix an arbitrary
$C\in C_{1}$, and let $X$ (resp. $Y$) be the point of $C_{2}$ closest (resp.
farthest) from $C$. We have $|CX|<|AB|$. Indeed, if $a=b$, then it is trivial
because $C=X$. Otherwise, it follows from the fact that $CX$ and $AB$ are
parallel and the orthogonal projections of $X$ and $C$ to the line $AB$ are
inside the segment $AB$, see Figure 1. The point $Y$ is the reflection of $C$
through the midpoint $O$ of the segment $AB$. Since $\measuredangle
ACB=\measuredangle AYB<90^{\circ}$, both $C$ and $Y$ are outside the ball of
center $O$ and radius $|AB|/2$. Consequently $|CY|>|AB|$. Therefore, there is
a point $D$ on $C_{2}$ with $|CD|=|AB|$. The pair $(C,D)$ is the desired
extension. ∎

Figure 1: Segment $AB$ is longer than $CX$ and shorter than $CY$.

######  Lemma 1.

Let $l_{1}$ and $l_{2}$ be two skew lines in $\mathbb{R}^{3}$. Suppose that
the set $S\subset\mathbb{R}^{3}$ is invariant under rotations about $l_{1}$
and $l_{2}$. Then $S=\varnothing$ or $S=\mathbb{R}^{3}$.

###### Proof.

We will use the following notation: $\mbox{E}^{+}$ is the group of all
orientation preserving isometries of $\mathbb{R}^{3}$. For any point $P$,
$\mbox{SO}(P)$ is the group of all orientation preserving isometries that do
not move the point $P$. For any line $l$, $\mbox{R}(l)$ is the group of all
rotations about $l$. The subgroup of translations of $\mathbb{R}^{3}$ is
denoted by T.

Suppose that $l_{1}$ and $l_{2}$ are two skew lines in $\mathbb{R}^{3}$ and
$S$ is invariant for the groups $\mbox{R}(l_{1})$ and $\mbox{R}(l_{2})$. Then
it is also invariant for the generated group
$\mbox{G}=\langle\mbox{R}(l_{1}),\mbox{R}(l_{2})\rangle$. We claim that
$\mbox{G}=\mbox{E}^{+}$, which is sufficient to show that either
$S=\varnothing$ or $S=\mathbb{R}^{3}$. Indeed, if for some points
$X,Y\in\mathbb{R}^{3}$, we have $X\in S$ but $Y\not\in S$, then $S$ is not
invariant for a translation by the vector $\overrightarrow{XY}$, a
contradiction.

Rotate the line $l_{1}$ about $l_{2}$. We get a hyperboloid $H_{1}$ which
contains $l_{1}$, and its axis is $l_{2}$. Similarly, rotate $l_{2}$ about
$l_{1}$. We get a hyperboloid $H_{2}$ which contains $l_{2}$, and its axis is
$l_{1}$. It follows, that the two hyperboloids intersect. Therefore, there is
a rotated copy $l_{1}^{\prime}$ of $l_{1}$, and a rotated copy
$l_{2}^{\prime}$ of $l_{2}$, that intersect in a point $P$. It is not hard to
see that a rotation about $l_{1}^{\prime}$ is equal to a conjugate of the
rotation about $l_{1}$ by the same angle with an element of $\mbox{R}(l_{2})$
that transforms $l_{1}$ into $l_{1}^{\prime}$. Hence
$\mbox{R}(l_{1}^{\prime})\subset\mbox{G}$. Similarly,
$\mbox{R}(l_{2}^{\prime})$ is a subgroup of G, and thus so is
$\mbox{G}^{\prime}=\langle\mbox{R}(l_{1}^{\prime}),\mbox{R}(l_{2}^{\prime})\rangle$.

We claim that $\mbox{G}^{\prime}=\mbox{SO}(P)$. The inclusion
$\mbox{G}^{\prime}\subseteq\mbox{SO}(P)$ is obvious. Take a unit sphere ${\cal
S}$ of center $P$. The groups $\mbox{R}(l_{1}^{\prime})$,
$\mbox{R}(l_{2}^{\prime})$, $\mbox{SO}(P)$ are orientation preserving
isometries of ${\cal S}$. It is known [3, Theorem 3.20] that each element of
$\mbox{SO}(P)$ is a rotation about a line through $P$. Let $Q_{1}$ (resp.
$Q_{2}$) be an intersection of the line $l_{1}^{\prime}$ (resp.
$l_{2}^{\prime}$) and ${\cal S}$. Consider the orbit $O(Q_{1})$ of the point
$Q_{1}$ under the action of $\mbox{G}^{\prime}$. It contains at least one
point $Q_{1}^{\prime}$ different from $Q_{1}$. As earlier, each element of
$\mbox{R}(PQ_{1}^{\prime})$ is a conjugate of the corresponding element of
$\mbox{R}(l_{1}^{\prime})$ with an element of $\mbox{G}^{\prime}$ that
transforms $Q_{1}$ into $Q_{1}^{\prime}$. Hence
$\mbox{R}(PQ_{1}^{\prime})\subset\mbox{G}^{\prime}$. If $\measuredangle
Q_{1}PQ_{1}^{\prime}<90^{\circ}$, then rotate $Q_{1}$ about the line
$PQ_{1}^{\prime}$ by $180^{\circ}$, let $Q_{1}^{\prime\prime}$ be the image.
Then $Q_{1}^{\prime\prime}$ is also in $O(Q_{1})$, and $\measuredangle
Q_{1}PQ_{1}^{\prime\prime}=2\measuredangle Q_{1}PQ_{1}^{\prime}$. By repeating
this procedure if necessary, we get a point $V_{1}\in O(Q_{1})$ such that
$\measuredangle Q_{1}PV_{1}\geq 90^{\circ}$. All elements $\mbox{G}^{\prime}$
are finite products of rotations about lines $l_{1}^{\prime}=Q_{1}P$ and
$l_{2}^{\prime}=Q_{2}P$. So, we can decrease all rotation angles continuously
to $0$. This shows that $\mbox{G}^{\prime}$ is connected, and we can find
$V_{1}^{\prime}\in O(Q_{1})$ such that $\measuredangle
Q_{1}PV_{1}^{\prime}=90^{\circ}$. But the rotations about two orthogonal axes,
$\mbox{R}(PQ_{1})$ and $\mbox{R}(PV_{1}^{\prime})$, act transitively on ${\cal
S}$. Therefore, we can get all rotation axes, and, consequently, all
rotations. This proves that $\mbox{G}^{\prime}=\mbox{SO}(P)$.

Consider the lines $l_{1}$ and $l_{1}^{\prime}$. Line $l_{1}$ does not contain
$P$, but $l_{1}^{\prime}$ does. Applying an appropriate element of
$\mbox{SO}(P)$, we can transform $l_{1}^{\prime}$ into a line
$l_{1}^{\prime\prime}$ which is parallel to $l_{1}$, but they are not
identical. Clearly, $\mbox{R}(l_{1}^{\prime\prime})\subset\mbox{G}$. Take the
composition of rotation about $l_{1}$ by angle $\alpha$ and a rotation about
$l_{1}^{\prime\prime}$ by angle $-\alpha$. The result is a translation
$\mbox{T}(\alpha)$. As $\alpha$ goes to $0$, the translation distance of
$\mbox{T}(\alpha)$ also goes to $0$ continuously. So, we can obtain a
translation by any given distance, but the direction changes. Taking its
conjugate with the elements of $\mbox{SO}(P)$, we can change the directions of
the translations arbitrarily, therefore, we can obtain any translation, so
$\mbox{T}\subset\mbox{G}$. Finally, $\mbox{E}^{+}=\mbox{T}\cdot\mbox{SO}(P)$,
therefore, $\mbox{G}=\mbox{E}^{+}$, as desired. ∎

Return to the proof of Theorem 2. Let
$S^{\prime}\subset\mathbb{R}^{3}\times\mathbb{R}^{3}$ be the following set of
pairs of points. Let the pair $(A,B)$ be in $S^{\prime}$, and take its closure
under the extension operation. In other words, the elements of $S^{\prime}$
are exactly those pairs that can be obtained by finitely many applications of
the extension operation, starting with the pair $(A,B)$. By Claim 1, all pairs
in $S^{\prime}$ are good pairs. Let $S\subset\mathbb{R}^{3}$ be the set of all
points included in at least one pair from $S^{\prime}$. Suppose that $(C,D)$
is an extension of $(A,B)$, and let $(C^{\prime},D^{\prime})$ be a rotation of
$(C,D)$ about the line $AB$, by an arbitrary angle. Then
$(C^{\prime},D^{\prime})$ is also an extension of $(A,B)$. Therefore, the set
$S$ is invariant under rotations about the lines $AB$ and $CD$. So we can
apply Lemma 1 to conclude that $S=\mathbb{R}^{3}$ or $S=\varnothing$. However,
$S$ is certainly not empty, as $A,B\in S$, therefore, $S=\mathbb{R}^{3}$.
Since all points of $S$ are either red or blue, we used only two colors for
the whole space. Now the following result of Erdős et al. guarantees that some
copy of $T$ is monochromatic, which completes the proof of 2 by contradiction.

######  Theorem 6 (Theorem 8 in [9]).

For every triangle $T$, we have $\mathbb{R}^{3}\xrightarrow{2}T$.

###  2.2 Hypercubes – proof of Theorem 4

For any
$A_{1}\subset\mathbb{R}^{d_{1}},\ldots,A_{n}\subset\mathbb{R}^{d_{n}}$, their
Cartesian product $A_{1}\times\cdots\times A_{n}$ is
$\left\\{(\mathbf{x}_{1},\ldots,\mathbf{x}_{n})\ :\ \mathbf{x}_{i}\in
A_{i}\right\\}\subset\mathbb{R}^{d_{1}+\cdots+d_{n}}$. If
$A_{1}=\cdots=A_{n}=A$, then $A_{1}\times\cdots\times A_{n}$ is the Cartesian
power of $A$ and it is denoted by $A^{n}$ for short. Let $I^{m}(a)$ be an
$m$-dimensional hypercube of side length $\sqrt{a}$ and $S_{m}(a)$ be an
$m$-vertex, that is, ($m-1$)-dimensional regular simplex of side length
$\sqrt{a}$. Considering the neighbors of a fixed vertex, it is easy to see
that $I^{m}(1)$ contains $S_{m}(2)$ as a subset. We also need the following
asymmetric generalization of the arrow notation: by
$C\smash{\xrightarrow[\smash{\raisebox{3.01385pt}{\tiny$\mathbf{MR}$}}]{}}(A,B)$,
we denote that claim that every coloring of $C$ contains either a
monochromatic copy of $A$, or a rainbow copy of $B$.

The main result of this section is the following statement.

######  Proposition 1.

For all $k,m\in\mathbb{N}$, there exists $n=n_{1}(k,m)$ such that
$I^{n}(1)\smash{\xrightarrow[\smash{\raisebox{3.01385pt}{\tiny$\mathbf{MR}$}}]{}}(I^{k}(2^{k}),I^{m}(2^{k}))$.

4 follows from the special case $k=m$, i.e., from the claim that
$I^{n}(1)\smash{\xrightarrow[\smash{\raisebox{3.01385pt}{\tiny$\mathbf{MR}$}}]{}}I^{m}(2^{m})$.
We prove 1 by induction on $k$ for any fixed $m$. First we prove the base
case.

######  Proposition 2.

For all $m\in\mathbb{N}$, there exists $n=n_{2}(m)$ such that
$I^{n}(1)\smash{\xrightarrow[\smash{\raisebox{3.01385pt}{\tiny$\mathbf{MR}$}}]{}}(I(2),I^{m}(2))$.

###### Proof.

We use the probabilistic method to prove the following technical statement.

For all $m,d\in\mathbb{N}$ such that $d>4^{m}$, the Cartesian power
$S_{d}^{m}(2)$ satisfies
$S_{d}^{m}(2)\smash{\xrightarrow[\smash{\raisebox{3.01385pt}{\tiny$\mathbf{MR}$}}]{}}(I(2),I^{m}(2))$.

This implies Proposition 2 since $S_{d}^{m}(2)\subset I^{md}(1)$. Consider an
arbitrary coloring of $S_{d}^{m}(2)$ and suppose that there is no
monochromatic copy of $I(2)$. Choose a pair of points from each of the $m$
simplices uniformly and independently at random. The Cartesian product of
these 2-point sets is a copy of $I^{m}(2)$. If this copy is not rainbow, then
some two of its vertices have the same color. We call it a bad event. The type
of the bad event is the pair of vertices of the same color. There are
$2^{m-1}(2^{m}-1)<4^{m}\leq d-1$ pairs of vertices of $I^{m}(2)$, so there are
at most $d-1$ types of bad events. Suppose that we fix a point from each of
the $m$ simplices, except for one, where we go over all of its vertices. We
get a copy of $S_{d}(2)$ in the product. Since no copy of $I(2)$ is
monochromatic, this copy of $S_{d}(2)$ is rainbow. This implies that the
probability of each bad event is at most $1/(d-1)$. Now the union bound
implies that a random copy of $I^{m}(2)$ is rainbow with a positive
probability, which completes the proof. ∎

Before the induction step, we need one more simple statement.

######  Proposition 3.

For all $r,m\in\mathbb{N}$, there exists $n=n_{3}(r,m)$ such that
$I^{n}(1)\xrightarrow{r}I^{m}(2)$.

###### Proof.

We prove the following technical statement by induction.

For all $n_{1},\dots,n_{m}$ such that
$n_{1}>r,n_{2}>n_{1}^{r},\dots,n_{m}>(n_{1}\cdot\dotsc\cdot n_{m-1})^{r}$, we
have $S_{n_{1}}(2)\times\dots\times S_{n_{m}}(2)\xrightarrow{r}I^{m}(2)$.

This implies the original statement since $S_{n_{1}}(2)\times\dots\times
S_{n_{m}}(2)\subset I^{n}(1)$ for $n=n_{1}+\dots+n_{m}$. The $m=1$ case is
trivial, since among $n_{1}>r$ vertices of $S_{n_{1}}(2)$ some two are of the
same color by the pigeonhole principle. To prove the induction step, we
consider $S_{n_{1}}(2)\times\dots\times S_{n_{m}}(2)$ as the union of $n_{m}$
‘layers’ of the form $S_{n_{1}}(2)\times\dots\times
S_{n_{m-1}}(2)\times\\{x\\}$, $x\in S_{n_{m}}(2)$. Since
$n_{m}>(n_{1}\cdot\dotsc\cdot n_{m-1})^{r}$, some two layers are colored
identically by the pigeonhole principle. Now the union of a monochromatic copy
of $I^{m-1}(2)$ from one of these layers, which must exist by the induction
hypothesis, with the corresponding set from the other layer form the desired
monochromatic copy of $I^{m}(2)$. ∎

Now we complete the proof of 1 for $k>1$ by induction. We show that the
desired statement holds for

$n=n_{1}(k,m)\coloneqq
n^{\prime}+n_{3}\big{(}r^{\prime},m^{\prime}\big{)},\mbox{ where
}n^{\prime}=n_{2}(m)2^{k-1},\ r^{\prime}=2^{n^{\prime}+k},\
m^{\prime}=n_{1}(k-1,m).$

Fix an arbitrary coloring of $I^{n}(1)$.

For each $\mathbf{x}\in I^{n-n^{\prime}}(1)$, we consider a ‘layer’
$\\{\mathbf{x}\\}\\!\times\\!I^{n^{\prime}}(1)\subset I^{n}(1)$. By
partitioning $n^{\prime}$ basic vectors of $I^{n^{\prime}}(1)$ into $n_{2}(m)$
groups of size $2^{k-1}$ and applying 2, we conclude that
$I^{n^{\prime}}(1)\smash{\xrightarrow[\smash{\raisebox{3.01385pt}{\tiny$\mathbf{MR}$}}]{}}(I(2^{k}),I^{m}(2^{k}))$.
If at least for one $\mathbf{x}\in I^{n-n^{\prime}}(1)$, the second
alternative holds, that is, there exists a rainbow copy of $I^{m}(2^{k})$ in
the layer $\\{\mathbf{x}\\}\\!\times\\!I^{n^{\prime}}(1)$, then we are done.
So we can assume without loss of generality that for all $\mathbf{x}\in
I^{n-n^{\prime}}(1)$, there exists a monochromatic copy of $I(2^{k})$ in the
layer $\\{\mathbf{x}\\}\\!\times\\!I^{n^{\prime}}(1)$.

Consider the auxiliary $r^{\prime}$-coloring of $I^{n-n^{\prime}}(1)$, where
each vertex $\mathbf{x}$ is colored according to which of the $r^{\prime}$
copies of $I(2^{k})$ in the layer
$\\{\mathbf{x}\\}\\!\times\\!I^{n^{\prime}}(1)$ is monochromatic. If the layer
contains several monochromatic copies of $I(2^{k})$, we pick one of them
arbitrarily. Now 3 implies that there exists a copy of $I^{m^{\prime}}(2)$
that is monochromatic under this auxiliary coloring. In terms of the original
coloring, this gives us two identically colored copies of $I^{m^{\prime}}(2)$
that form a ‘prism’ $I^{m^{\prime}}(2)\times I(2^{k})$.

Applying the induction hypothesis to the base of this prism, we find either a
monochromatic copy of $I^{k-1}(2\cdot 2^{k-1})$, or a rainbow copy of
$I^{m}(2\cdot 2^{k-1})$ there. In the latter case, we are done immediately. In
the former case, the union of this monochromatic copy of $I^{k-1}(2^{k})$ in
the first level of the prism with the corresponding identically colored set
from the second level form the desired monochromatic copy of $I^{k}(2^{k})$.
This completes the proof of 1.

##  3 Maximum norm

###  3.1 Hales–Jewett theorem

In this subsection, we discuss the Hales–Jewett theorem, which is a central
result in Ramsey and also the main tool of this section. Informally speaking,
it states that a sufficiently high dimensional grid, colored with a given
number of colors, contains a given dimensional, monochromatic subgrid of a
certain type, described below.

For $k,n\in\mathbb{N}$, let $\tau$ be an element of $([k]\cup\\{*\\})^{n}$
with at least one $*$-coordinate, where $*$ is an abstract symbol that we call
a variable. We define a combinatorial line corresponding to $\tau$ as a subset
$\\{\tau(i):i\in[k]\\}\subset[k]^{n}$, where $\tau(i)$ is an element of
$[k]^{n}$ that we obtain by replacing all the variable coordinates of $\tau$
by $i$. More generally, let $\tau$ be the element of
$([k]\cup\\{*_{1},\dots,*_{m}\\})^{n}$ with at least one variable coordinate
of each of the $m$ types. We define an $m$-dimensional combinatorial subspace
corresponding to $\tau$ as a subset
$\\{\tau(i_{1},\dots,i_{m}):i_{1},\dots,i_{m}\in[k]\\}\subset[k]^{n}$, where
$\tau(i_{1},\dots,i_{m})$ is an element of $[k]^{n}$ that we obtain by
replacing all the variable coordinates of $\tau$ of the $j$-th type by $i_{j}$
for each $j\in[m]$. In this notation, the multidimensional Hales–Jewett
theorem is the following statement.

######  Theorem 7 (Multidimensional Hales–Jewett theorem).

For all $k,r,m\in\mathbb{N}$, there is a sufficiently large $n\in\mathbb{N}$
such that for every $r$-coloring of $[k]^{n}$, there exists a monochromatic
$m$-dimensional combinatorial subspace.

Hales and Jewett [24] proved this theorem for $m=1$ in 1963, while the
multidimensional generalization is due to Graham and Rothschild [22].
Furstenberg and Katznelson[18] proved a density versions of this theorem using
ergodic theory, while in 2009, the Polymath Project [36] developed a
combinatorial proof.

For the canonical version, if $k>2$, then it is not true that for every
coloring of $[k]^{n}$, there exists either a monochromatic or a rainbow
$m$-dimensional combinatorial subspace even if $n$ is arbitrary large. Indeed,
let $\sim$ be an equivalence relation on $[k]$. Consider a coloring $c_{\sim}$
of $[k]^{n}$ such that for all $\mathbf{x}=(x_{1},\dots,x_{n})$ and
$\mathbf{y}=(y_{1},\dots,y_{n})$ in $[k]^{n}$, we have

$c_{\sim}(\mathbf{x})=c_{\sim}(\mathbf{y})\mbox{ \ if and only if \ }x_{1}\sim
y_{1},\dots,x_{n}\sim y_{n}.$

Observe that the induced coloring on every $m$-dimensional combinatorial
subspace of $[k]^{n}$ coincides with the coloring $c_{\sim}$ on $[k]^{m}$
under a natural isomorphism. Therefore, no $m$-dimensional combinatorial
subspace is monochromatic or rainbow, unless $\sim$ is a trivial equivalence
relation, that is, the elements of $[k]$ are either pairwise equivalent or
pairwise non-equivalent. However, Prömel and Voigt [37, Theorem C.7] showed
that one can always find a combinatorial subspace colored according to some
$c_{\sim}$, see also [35].

######  Theorem 8 (Canonical Hales–Jewett theorem).

For all $k,m\in\mathbb{N}$, there is a sufficiently large $n\in\mathbb{N}$
such that for every coloring of $[k]^{n}$, there exists an $m$-dimensional
combinatorial subspace such that the induced coloring on it coincides with
$c_{\sim}$ for some equivalence relation $\sim$ on $[k]$.

###  3.2 Proof of Theorem 5

In this section, we deduce 5 from the Canonical Hales–Jewett theorem. Observe
that our finite $M\subset\mathbb{R}^{d}$ is a subset of a Cartesian power
$A^{d}$ for some finite $A\subset\mathbb{R}$. Indeed, let $A$ be the union of
the projections of $M$ onto each of the $d$ coordinate axes. Hence, it is
sufficient to prove 5 only for $A^{d}$ playing the role of $M$.

Label the elements of $A$ by $a_{1},\dots,a_{s}$ in ascending order. By adding
between every two of its consecutive elements sufficiently many equally spaced
intermediate points, we obtain a set $B=\\{b_{1},\dots,b_{k}\\}\supset A$ such
that

$\max_{1\leq i<k}(b_{i+1}-b_{i})\leq\min_{1\leq j<s}(a_{j+1}-a_{j}),$ (1)

where the elements of $B$ are also labeled in ascending order, see Figure 2.

Let $n$ be from the statement of 8 applied to our $k$ and $m=d+\lceil
d\log_{2}s\rceil$. In what follows, we prove that every coloring of $B^{n}$
contains either a monochromatic or a rainbow $\ell_{\infty}$-isometric copy of
$A^{d}$, which implies that
$\mathbb{R}_{\infty}^{n}\smash{\xrightarrow[\smash{\raisebox{3.01385pt}{\tiny$\mathbf{MR}$}}]{}}A^{d}$.
Consider an arbitrary coloring of $B^{n}$. A natural bijection $\varphi$
between $[k]^{n}$ and $B^{n}$ defined by

$\varphi(x_{1},\dots,x_{n})=(b_{x_{1}},\dots,b_{x_{n}})\mbox{ for all
}(x_{1},\dots,x_{n})\in[k]^{n}$

translates that coloring into a coloring of $[k]^{n}$. Now 8 implies that the
$m$-dimensional combinatorial subspace corresponding to some
$\tau\in([k]\cup\\{*_{1},\dots,*_{m}\\})^{n}$ is colored according to
$c_{\sim}$ for some equivalence relation $\sim$ on $[k]$.

If all the elements of $[k]$ are pairwise equivalent with respect to $\sim$,
i.e., if the combinatorial space is monochromatic, then it is easy to complete
the proof. Indeed, consider the subset of our combinatorial space where the
last $m-d$ variables stay fixed, say, $*_{d+1}=\dots=*_{m}=1$, while the first
$d$ variables range over the indices of the elements of $A$ inside $B$. It is
not hard to see that the image of this subset under $\varphi$ is the desired
monochromatic $\ell_{\infty}$-isometric copy of $A^{d}$ inside $B^{n}$. More
precisely, let $1=i_{1}<\dots<i_{s}=k$ be a sequence such that
$a_{j}=b_{i_{j}}$ for all $j\in[s]$, and put
$I=\\{i_{1},\dots,i_{s}\\}\subset[k]$, see Figure 2. Let $\mathcal{I}_{1}$ be
the set of all $s^{d}$ points of $[k]^{m}$ such that their first $d$
coordinates belong to $I$, while the remaining $m-d$ coordinates are equal to
$1$. Note that $\tau(\mathcal{I}_{1})\subset[k]^{n}$ is a subset of our
combinatorial subspace, and thus it is monochromatic. Moreover,
$\varphi(\tau((\mathcal{I}_{1}))\subset B^{n}$ is an $\ell_{\infty}$-isometric
copy of $A^{d}$, because the $\ell_{\infty}$-norm is independent of the
arguments’ multiplicities, i.e., the number of $*_{r}$-coordinates of $\tau$
for all $r\in[d]$, as desired.

Figure 2: For $A=\\{0,16,20,26\\}$, we can take
$B=\\{0,4,8,12,16,20,23,26\\},\ I=\\{1,5,6,8\\}$.

Otherwise, there exists $1\leq i^{\prime}<k$ such that $i^{\prime}\not\sim
i^{\prime}+1$. Let $\sigma$ be an arbitrary injection from $I^{d}$ to
$\\{i^{\prime},i^{\prime}+1\\}^{m-d}$, where $I$ is from the previous
paragraph. Such an injection exists because $|I^{d}|=s^{d}\leq
2^{m-d}=|\\{i^{\prime},i^{\prime}+1\\}^{m-d}|$ by our choice of $m$. Consider
$\mathcal{I}_{2}=\\{(\mathbf{x},\sigma(\mathbf{x})):\mathbf{x}\in
I^{d}\\}\subset[k]^{m}$. Note that every two different elements of
$\mathcal{I}_{2}$ have at least one non-equivalent coordinate, and thus the
set $\tau(\mathcal{I}_{2})\subset[k]^{n}$ is rainbow. Moreover, it is not hard
to check that $\varphi(\tau(\mathcal{I}_{2}))\subset B^{n}$ is an
$\ell_{\infty}$-isometric copy of $A^{d}$. Indeed, in the previous paragraph,
we have already seen this when the last $m-d$ variable coordinates were fixed.
Now they are not fixed, but vary between $i^{\prime}$ and $i^{\prime}+1$.
However, (1) implies that $b_{i^{\prime}+1}-b_{i^{\prime}}$ is not larger than
the smallest $\ell_{\infty}$-distance between different elements of $A^{d}$.
Therefore, these additional variable coordinates do not change all the
$\ell_{\infty}$-distances. This completes the proof 5.

##  4 Concluding remarks

Canonical types. For any finite set $A$, let
$\mathbb{R}^{n}\smash{\xrightarrow[\smash{\raisebox{3.01385pt}{\tiny$\mathbf{MR}$}}]{r}}A$
denote the claim that every $r$-coloring of $\mathbb{R}^{n}$ contains a
monochromatic or a rainbow copy of $A$. Clearly, if
$\mathbb{R}^{n}\smash{\xrightarrow[\smash{\raisebox{3.01385pt}{\tiny$\mathbf{MR}$}}]{r}}A$
then also
$\mathbb{R}^{n+1}\smash{\xrightarrow[\smash{\raisebox{3.01385pt}{\tiny$\mathbf{MR}$}}]{r}}A$
and
$\mathbb{R}^{n}\smash{\xrightarrow[\smash{\raisebox{3.01385pt}{\tiny$\mathbf{MR}$}}]{r-1}}A$.
Therefore, we can distinguish the following three types of sets, see Figure 3.

  1. 1.

There is $r$ such that
$\mathbb{R}^{n}\smash{\not\xrightarrow[\smash{\raisebox{3.01385pt}{\tiny$\mathbf{MR}$}}]{r}}A$
for every $n$.

  2. 2.

There is $n$ such that
$\mathbb{R}^{n}\smash{\xrightarrow[\smash{\raisebox{3.01385pt}{\tiny$\mathbf{MR}$}}]{r}}A$
for every $r$.

  3. 3.

None of the above, i.e., for every $r$ there is $n$ with
$\mathbb{R}^{n}\smash{\xrightarrow[\smash{\raisebox{3.01385pt}{\tiny$\mathbf{MR}$}}]{r}}A$,
and for every $n$ there is $r$ with
$\mathbb{R}^{n}\smash{\not\xrightarrow[\smash{\raisebox{3.01385pt}{\tiny$\mathbf{MR}$}}]{r}}A$.

Figure 3: All three possible types of finite sets.

It is clear that Ramsey sets cannot be of type 1, and it is natural to ask
whether they are all of type 2. In fact, we cannot fully answer this question
even if the Ramsey set $A$ is a rectangle or an obtuse triangle. Our 4 implies
that rectangles with side lengths $a$ and $b$ are of type 2, provided that
$(b/a)^{2}$ is rational, but we could not determine a type of the remaining
rectangles. As we mentioned in the Introduction, Cheng and Xu [4] proved that
the obtuse triangles that are not ‘too flat’ are of type 2. The type of ’flat’
obtuse triangles remains uncertain. We do not have an example of a Ramsey set
of type 3.

For non-Ramsey sets we can not rule out that all three types are possible. As
it was observed in [4], a classical spherical $3$-coloring of $\mathbb{R}^{n}$
without monochromatic copies of a $4$-term unit arithmetic progression $l_{4}$
from [9] cannot contain a rainbow copy either, so $l_{4}$ is of type 1. We are
not aware of any non-Ramsey set of other types. A $3$-term unit arithmetic
progression $l_{3}$ might be natural candidate for type 2 (with $n$ being $2$
or $3$), since it was recently shown in [5] that
$\mathbb{R}^{2}\xrightarrow{2}l_{3}$.

Tightness of 2. A classical $2$-coloring of the plane with color-alternating
strips of heights $\sqrt{3}/2$ does not contain monochromatic copies of a unit
equilateral triangle $\triangle$. Hence, it is not true that
$\mathbb{R}^{2}\smash{\xrightarrow[\smash{\raisebox{3.01385pt}{\tiny$\mathbf{MR}$}}]{}}\triangle$.
Are there other triangles $T$ for which the dimension $3$ in 2 is also tight?
It was conjectured in [11, Conjecture 3] that $\mathbb{R}^{2}\xrightarrow{2}T$
for every non-equilateral triangle $T$. If true, this would be an evidence
supporting that
$\mathbb{R}^{2}\smash{\xrightarrow[\smash{\raisebox{3.01385pt}{\tiny$\mathbf{MR}$}}]{}}T$
might hold for all non-equilateral triangles $T$, and thus our result is not
tight. However, during the last 50 yeas, that conjecture was verified only for
a few special families of triangles, see [34] and the references therein.

Quantitative bounds in 4. What is the minimum $n=n(m)$ such that
$\mathbb{R}^{n}\smash{\xrightarrow[\smash{\raisebox{3.01385pt}{\tiny$\mathbf{MR}$}}]{}}I^{m}$?
A careful analysis of our proof leads to the upper bound
$n(m)=\exp_{2}^{2m-1}\big{(}O(m)\big{)}$, where $\exp_{2}^{1}(x)=2^{x}$ and
$\exp_{2}^{k+1}(x)=2^{\exp_{2}^{k}(x)}$ for all $k>1$. However, our argument
is very wasteful in a sense that all the auxiliary propositions are stated for
the hypercubes, but in fact we are working only with their small subsets:
Cartesian product of simplices. Taking this into account, we can improve the
upper bound to $n(m)=\exp_{2}^{4}\big{(}O(m)\big{)}$, which is still a
quadruple exponent. From the other direction, we know that $n(m)=\Omega(m\log
m)$, since for smaller $n$, there exists a coloring of $\mathbb{R}^{n}$ with
no monochromatic isometric copies of $I^{m}$ that uses less than $2^{m}$
colors, see [38].

The case $m=2$ is of a particular interest here. We can show that
$K_{876}\square K_{123}\square K_{4}\square
K_{2}\smash{\xrightarrow[\smash{\raisebox{3.01385pt}{\tiny$\mathbf{MR}$}}]{}}K_{2}\square
K_{2}$, where $\square$ stands for the Cartesian product of these cliques. A
straightforward translation of this purely graph theoretical statement into
geometric language implies that $n(2)\leq 875+122+3+1=1001$. We do not know
what is the best upper bound on $n(2)$ that can be derived from [4], but it
might be of the same order. From the other direction, we do not even know if
$n(2)>3$.

Max-norm Ramsey theory. Though our 5 is rather general, the quantitative
bounds on the minimum dimension $n=n(M)$ such that
$\mathbb{R}_{\infty}^{n}\smash{\xrightarrow[\smash{\raisebox{3.01385pt}{\tiny$\mathbf{MR}$}}]{}}M$
it generates are probably extremely far from being tight. For instance, if $M$
is a rectangle with side lengths $a<b$, then the number of auxiliary points on
Figure 2, as well as the resulting upper bound, will depend on the fraction
$b/a$. We found an ad hoc argument for this case in the spirit of [4] which
leads to the upper bound $O(b/a)$, i.e., this dependence is at most linear.
Does there exist a universal constant $n$ such that
$\mathbb{R}_{\infty}^{n}\smash{\xrightarrow[\smash{\raisebox{3.01385pt}{\tiny$\mathbf{MR}$}}]{}}R$
for every rectangle $R$?

Manhattan Ramsey theory. Let $\mathbb{R}_{1}^{n}$ be the $n$-dimensional space
equipped with the Manhattan $\ell_{1}$-norm. It is not hard to show that for
every $r$ and a finite $M\subset\mathbb{R}^{d}$, there exists $n$ such that
$\mathbb{R}_{1}^{n}\xrightarrow{r}M$, see [31]. Our 1 has the following
canonical corollary: for every finite $M\subset\mathbb{Q}^{d}$, there exists
$n$ such that
$\mathbb{R}_{1}^{n}\smash{\xrightarrow[\smash{\raisebox{3.01385pt}{\tiny$\mathbf{MR}$}}]{}}M$.
Indeed, it is sufficient to note that a path of $m$ pairwise orthogonal edges
in the hypercube is an $\ell_{1}$-isometric copy of an arithmetic progression.
Besides that, every ‘rational’ $M$ can be embedded into a Cartesian power of
this progression after a proper scaling. We wonder, if the same conclusion
holds for ‘non-rational’ sets $M$ as well, e.g. for all rectangles.

Acknowledgements. We are very grateful to Endre Szabó and Balázs Csikós for
their help in the presentation of Lemma 1. Arsenii Sagdeev thanks Alexander
Golovanov for the helpful discussion of Lemma 1.

Panna Gehér was supported by the Lendület Programme of the Hungarian Academy
of Sciences – grant number LP2021-1/2021 and by the Thematic Excellence
Program TKP2021-NKTA-62 of the National Research, Development and Innovation
Office. Panna Gehér and Géza Tóth were supported by the National Research,
Development and Innovation Office, NKFIH, K-131529. Arsenii Sagdeev and Géza
Tóth were supported by ERC Advanced Grant ‘GeoScape’ No. 882971.

## References

  * [1] Axenovich, M., Jamison, R. E. (2005). Canonical pattern Ramsey numbers. Graphs Combin., 21, 145–160.
  * [2] Cantwell, K. (2007). All regular polytopes are Ramsey. J. Combin. Theory Ser. A, 114(3), 555–562.
  * [3] Cederberg, J. (1989). A course in modern geometries. Springer Science & Business Media.
  * [4] Cheng, X., Xu, Z. (2023). Euclidean Gallai-Ramsey for various configurations. preprint arXiv:2305.18218.
  * [5] Currier, G., Moore, K., Yip, C. H. (2024). Any two-coloring of the plane contains monochromatic 3-term arithmetic progressions. preprint arXiv:2402.14197.
  * [6] Davies, J. (2023). Chromatic number of spacetime. preprint arXiv:2308.16885.
  * [7] De Grey, A. D. (2018). The chromatic number of the plane is at least $5$. Geombinatorics, 28(1), 18–31.
  * [8] Erdős, P., Graham, R. L. (1980). Old and new problems and results in combinatorial number theory. (Vol. 28). L’Enseignement Math, Monographie, Geneva.
  * [9] Erdős, P., Graham, R. L., Montgomery, P., Rothschild, B. L., Spencer, J., Straus, E. G. (1973). Euclidean Ramsey theorems I. J. Combin. Theory Ser. A, 14(3), 341–363.
  * [10] Erdős, P., Graham, R. L., Montgomery, P., Rothschild, B. L., Spencer, J., Straus, E. G. (1975). Euclidean Ramsey theorems II. Colloq. Math. Soc. János Bolyai, Infinite and Finite Sets, Keszthely, Hungary and North-Holland, Amsterdam, 520–557.
  * [11] Erdős, P., Graham, R. L., Montgomery, P., Rothschild, B. L., Spencer, J., Straus, E. G. (1975). Euclidean Ramsey theorems III. Colloq. Math. Soc. János Bolyai, Infinite and Finite Sets, Keszthely, Hungary and North-Holland, Amsterdam, 559–583.
  * [12] Erdős, P., Rado, R. (1950). A combinatorial theorem. J. Lond. Math. Soc., 1(4), 249–255.
  * [13] Exoo, G., Fisher, D., Ismailescu, D. (2021). The chromatic number of the Minkowski plane – the regular polygon case. Geombinatorics, 31, 49–67.
  * [14] Exoo, G., Ismailescu, D. (2020). The chromatic number of the plane is at least $5$: A new proof. Discrete Comput. Geom., 64(1), 216–226.
  * [15] Frankl, N., Gehér, P., Sagdeev, A., Tóth, G. (2023). Monochromatic infinite sets in Minkowski planes. preprint arXiv:2308.08840.
  * [16] Frankl, N., Kupavskii, A., Sagdeev, A. (2024). Max-norm Ramsey Theory. European J. Combin., 118, 103918.
  * [17] Frankl, P., Rödl, V. (1990). A partition property of simplices in Euclidean space. J. Amer. Math. Soc., 3(1), 1–7.
  * [18] Furstenberg, H., Katznelson, Y. (1991). A density version of the Hales–Jewett Theorem. Journal d’Analyse Mathématique, 57(1), 64–119.
  * [19] Gehér, P. (2023). Note on the chromatic number of Minkowski planes: the regular polygon case. Studia Sci. Math. Hungar., 60(1), 27–42.
  * [20] Gasarch, W., Kruskal, C., Parrish, A. (2012). Van der Waerden’s theorem: Variants and applications.
  * [21] Girão, A. (2020) A Canonical Polynomial Van der Waerden’s Theorem. preprint arXiv:2004.07766.
  * [22] Graham, R. L., Rothschild, B. L. (1971). Ramsey’s theorem for $n$-parameter sets. Trans. Amer. Math. Soc., 159, 257–292.
  * [23] Graham, R. L., Rothschild, B. L., Spencer, J. H. (1991). Ramsey theory. (Vol. 20). John Wiley & Sons.
  * [24] Hales, A. W., Jewett, R. I. (1963). Regularity and positional games. Trans. Amer. Math. Soc., 106(2), 222–229.
  * [25] Jamison, R. E., Jiang, T., Ling, A. C. H. (2003). Constrained Ramsey numbers of graphs. J. Graph Theory 42, 1–16.
  * [26] Jiang, T., Mubayi, D. (2000). New upper bounds for a canonical Ramsey problem. Combinatorica, 20(1), 141–146.
  * [27] Kříž, I. (1991). Permutation groups in Euclidean Ramsey theory. Proc. Amer. Math. Soc., 112(3), 899–907.
  * [28] Kirova, V., Sagdeev, A. (2023). Two-colorings of normed spaces without long monochromatic unit arithmetic progressions. SIAM J. Discrete Math., 37(2), 718–732.
  * [29] Kupavskiy, A. (2011). On the chromatic number of $R^{n}$ with an arbitrary norm. Discrete Math., 311(6), 437–440.
  * [30] Kupavskii, A., Sagdeev, A. (2021). All finite sets are Ramsey in the maximum norm. Forum Math., Sigma, 9, e55.
  * [31] Kupavskii, A., Sagdeev, A., Zakharov, D. (2022). Cutting corners. preprint arXiv:2211.17150.
  * [32] Leader, I., Russell, P.A., Walters, M. (2012). Transitive sets in Euclidean Ramsey theory. J. Combin. Theory Ser. A, 119(2), 382–396.
  * [33] Mao, Y., Oeki, K., Wang, Z. (2022). Euclidean Gallai-Ramsey Theory. preprint arXiv:2209.13247.
  * [34] Natalchenko, A., Sagdeev, A. (2023). Monochromatic triangles in the max-norm plane. preprint arXiv:2302.09972.
  * [35] Nešetrřil, J., Prömel, H. J., Rödl, V., Voigt, B. (1985). Canonizing ordering theorems for Hales Jewett structures. J. Combin. Theory Ser. A, 40(2), 394–408.
  * [36] Polymath, D. H. J. (2012). A new proof of the density Hales–Jewett theorem. Ann. of Math., 1283–1327.
  * [37] Prömel, H. J., Voigt, B. (1983). Canonical partition theorems for parameter sets. J. Combin. Theory Ser. A, 35(3), 309–327.
  * [38] Prosanov, R. I. (2018). Upper Bounds for the Chromatic Numbers of Euclidean Spaces with Forbidden Ramsey Sets. Math. Notes, 103, 243–250.
  * [39] Raigorodskii, A. M. (2004). On the chromatic number of a space with the metric $\ell_{p}$. Russian Math. Surveys, 59(5), 973.
  * [40] Soifer, A. (2009). The mathematical coloring book. Springer-Verlag, New York.
  * [41] Voronov, V. (2023). The chromatic number of the plane with an interval of forbidden distances is at least 7. preprint arXiv:2304.10163.
  * [42] Van der Waerden, B. L. (1927). Beweis einer baudetschen vermutung. Nieuw Arch. Wiskunde, 15, 212–216.
  * [43] Wagner, P. (2006). An upper bound for constrained Ramsey numbers. Combin. Probab. Comput., 15, 619–626.


∎

11institutetext: Cai-Yun Zhao 22institutetext: Yi Zhang 33institutetext: Hui-
Li Han 44institutetext: Ting-Yun Shi 55institutetext: State Key Laboratory of
Magnetic Resonance and Atomic and Molecular Physics, Wuhan Institute of
Physics and Mathematics, Innovation Academy for Precision Measurement Science
and Technology, Chinese Academy of Sciences, Wuhan 430071, P. R. China
66institutetext: Cai-Yun Zhao 77institutetext: Yi Zhang 88institutetext:
University of Chinese Academy of Sciences, 100049, Beijing, P. R. China
99institutetext: Hui-Li Han 1010institutetext: 1010email<EMAIL_ADDRESS>
# Hyperspherical approach to atom–dimer collisions with the Jacobi boundary
condition

Cai-Yun Zhao  Yi Zhang  Hui-Li Han  Ting-Yun Shi

(Received: date / Accepted: date)

###### Abstract

In this study, we investigate atom–dimer scattering within the framework of
hyperspherical coordinates. The coupled-channel Schrödinger equation is solved
using the R-matrix propagation technique combined with the smooth variable
discretization method. In the matching procedure, the asymptotic wave
functions are expressed in the rotated Jacobi coordinates. We apply this
approach to the elastic scattering 3He(T$\uparrow$) + 4He2 and H$\uparrow$
+LiH$\uparrow$ processes for testing. The convergence of the scattering length
as a function of the propagation distance is studied. We find that the method
is reliable and can provide considerable savings over previous propagators, so
it is suitable for solving the atom–dimer scattering problem for important
quantities such as the phase shift, cross section and scattering length.

##  1 Introduction

Studies of three-body collision processes have attracted tremendous attention
due to their substantial relevance in the rapidly growing field of cold and
ultracold atomic gases PhysRevLett.90.053202 ; PhysRevLett.103.083202 ;
RevModPhys.89.035006 ; Giannakeas2017Van . In such systems, elastic
atom–molecule collisions are crucial for determining the dynamics of ultracold
atom–molecule mixtures at the mean field level, and inelastic atom–molecule
collisions have a large impact on the lifetime of Feshbach molecules.

Weakly attracted three-body systems such as helium trimer and mixed 4He-4He-A
(A is another atom) systems are very interesting and important as they give us
an opportunity to study the Efimov states in the realistic systems 2014Imaging
; PhysRevA.79.024501 ; Wumengshan2014 . Interest in the Efimov states and
other universal binding properties of such systems has been significantly
investigated, and giant Efimov trimer has been detected in helium gas
Kunitski2015He . The scattering processes at ultralow energies are even more
interesting due to their relevance for the lifetime and stability of gas
samples. A few works have addressed the ultracold atom–molecule problem in
these realistic systems. For instance, ultracold collisions of 3,4He atoms
with 4He2 have been studied within the adiabatic hyperspherical representation
by Refs. PhysRevA.78.062701 ; PhysRevA.83.032703 . The Faddeev differential
equations have also been extensively used in these systems 2004Binding ;
2006Scattering ; Kolganova2009Ultracold . In addition to the well-studied
elastic 4He(3He)+4He2 scattering, the spin-stretched case of H atom scattering
from XH (X is an alkali atom) has been investigated using the method of
hyperspherical coordinates PhysRevA.83.032703 . Recently, atom–dimer exchanges
and dissociation reaction rates have been predicted for different combinations
of two 4He atoms and one of the alkaline species among 6Li, 7Li, and 23Na
using the Faddeev formalism PhysRevA.102.062814 . On the other hand, it is
known that there exist many similarities between spin-polarized tritium
(T$\uparrow$) and 4He atoms. The bulk T$\uparrow$ remains liquid in the limit
of zero temperature and behaves much like liquid 4He and therefore constitutes
a second example of a bosonic superfluid. The bound states of mixed
T$\uparrow^{4}$He2 clusters were studied in Refs. PhysRevLett.113.253401 ;
Hiroya2014A ; doi:10.1063/1.3530837  and were found to possess one weakly
bound state, which is by far the most weakly bound system. For this system, no
scattering observables are available in the literature, which is of
fundamental importance for current experiments.

The hyperspherical adiabatic (HA) expansion method has been proven to be an
efficient tool in studying few-atom systems CDLIN1995 . For bound states, HA
expansion shows particularly fast convergence for atom–atom interactions
Wumengshan2014 ; doi:10.1063/1.3451073 ; PhysRevA.79.024501 ;
PhysRevA.84.014501 . On the other hand, the method has also been extensively
used to describe few-atom systems in the ultracold collision regime
PhysRevLett.90.053202 ; PhysRevA.78.062701 ; PhysRevA.83.032703 . The
convergence problem of the HA method appears for scattering states,
particularly in the description of ultracold atom–dimer collisions. Since the
asymptotic structure for atom–dimer scattering is that one particle moves
relative to the center of mass of the two-body bound system, the correct
boundary condition for the structure with the HA basis is achieved only at
$\rho\rightarrow\infty$, which requires a very large number of hyperradial
functions in the solutions and long-range propagation PhysRevLett.103.090402 .
To overcome the convergence problem, Refs. PhysRevLett.103.090402 ;
PhysRevA.83.022705 ; 2008Three  introduced a method to compute the phase shift
from two integral relations that involve only the internal part of the wave
function. The convergence of the procedure has been demonstrated to be as fast
as for bound states.

An alternative method for addressing this problem is using asymptotic
solutions expressed in Jacobi coordinates. This idea has been applied to treat
rearrangement collisions by several quantum chemistry groups since 1980
doi:10.1063/1.476337 . In the calculations of Refs. DUEBILLING1980254 ;
BONDI1982570 ; doi:10.1021/j150644a017 , the probabilities were found to
exhibit no oscillations as a function of the matching distance. Then, in
treating the collision-induced dissociation problem, Refs.
doi:10.1063/1.1573186 ; Parker  used a mixed boundary condition scheme, in
which the asymptotic bound solutions were expressed in Jacobi coordinates and
the continuum solutions were expressed in hyperspherical coordinates. This
method significantly decreases the amplitude of the oscillations and improves
the convergence as a function of the distance. In atomic physics, the
hyperspherical close-coupling method, which uses Jacobi asymptotic solutions,
has been used to calculate the elastic and positronium formation cross section
for electron and positron collisions with atomic hydrogen 1994 ; Zhou_1995 ;
PhysRevA.50.232  and to study the photoionization cross section spectra of the
two-electron system Zhou_19931 ; Zhou_19932 . Zhao et al. PhysRevA.62.042706
also used the hyperspherical close-coupling method to investigate the charge
transfer process $A^{+}+B\rightarrow A+B^{+}$. In the calculations of the low-
energy collision of Coulomb three-body systems, Refs. PhysRevA.92.032713 ;
doi:10.1063/1.476337  used the hyperspherical elliptic coordinates method.
Their two-dimensional matching procedure also used the asymptotic wave
function expressed in the mass-scaled Jacobi coordinates. For the ultracold
atom–dimer elastic scattering, the collision quantities reach the threshold
regime only at collision energies at the $nk$ level, leading to longer
propagation than in the systems described above. Thus, there is a great need
to project numerical wave functions onto asymptotic solutions expressed in
Jacobi coordinates to study the ultracold atom–dimer scattering process.

In this work, we present an efficient method for investigating atom–dimer
scattering within the framework of hyperspherical coordinates. The
nonadiabatic coupling between the hyperradius and hyperangular variables is
treated with the slow-variable discretization (SVD) method Tolstikhin_1996  in
combination with the R-matrix propagation technique doi:10.1063/1.432836 ;
PhysRevA.84.052721 . In the matching procedure, the asymptotic wave functions
are expressed in the rotated Jacobi coordinates. We perform test calculations
on the 3He + 4He2 and H$\uparrow$ \+ H$\uparrow$Li systems, which represent
two different kinds of asymptotic structures. The two systems have been
studied previously with the asymptotic wave function expressed in the
hyperspherical coordinates, and the propagation distance is at least 5000 a.u.
PhysRevA.83.032703 . Thus, these systems are good examples to illustrate our
new approach. We also investigate the T$\uparrow$ \+ 4He2 elastic scattering
in $J^{\Pi}=0^{+}$ symmetries and provide the scattering length value for
T$\uparrow$ atom scattering from the 4He2 dimer.

The organization of this paper is as follows: Sec. II describes the
theoretical approach. In Sec. III, we discuss the results and analyses of the
systems under study. Finally, we conclude and summarize our work in Sec. IV.

##  2 Theoretical formalism

In this work, we consider a process where a particle hits a bound two-body
system. We assume the incident energy to be below the breakup threshold for
the three particles, and only the channels approaching the two-body bound
state need to be considered.

We use $m_{\tau}$ ($\tau$=A, B, C) to represent the mass of three atoms and
use $\mathbf{x_{\tau}}$ to represent the column vector relative to the origin.
In the center-of-mass frame, six coordinates are needed to describe the three-
particle system. The Jacobi coordinates describing the relative motion are
defined as

$\begin{array}[]{l}\mathbf{\rho}_{2\tau}=\mathbf{x}_{\tau}-\frac{m_{\tau+1}\mathbf{x}_{\tau+1}+m_{\tau+2}\mathbf{x}_{\tau+2}}{m_{\tau+1}+m_{\tau+2}},\\\
\mathbf{\rho}_{1\tau}=\mathbf{x}_{\tau+2}-\mathbf{x}_{\tau+1},\end{array}$ (1)

where $\tau,\tau+1,\tau+2$ are any cyclic permutations of A, B, and C. This is
illustrated in Fig. 1. In addition, $\mathbf{\xi_{1\tau}}$ and
$\mathbf{\xi_{2\tau}}$ are the corresponding mass-scaled Jacobi coordinates:

$\mathbf{\xi}_{2\tau}=\sqrt{\frac{\mu_{2}}{\mu}}\mathbf{\rho}_{2\tau},\quad\mathbf{\xi}_{1\tau}=\sqrt{\frac{\mu_{1}}{\mu}}\mathbf{\rho}_{1\tau},$
(2)

where $\mu_{1}=\frac{m_{\tau+1}m_{\tau+2}}{m_{\tau+1}+m_{\tau+2}}$,
$\mu_{2}=\frac{m_{\tau}(m_{\tau+1}m_{\tau+2})}{m_{\tau}+m_{\tau+1}+m_{\tau+2}}$
and
$\mu=\left[\frac{m_{\tau}m_{\tau+1}m_{\tau+2}}{m_{\tau}+m_{\tau+1}+m_{\tau+2}}\right]^{1/2}$.
Different sets of mass-scaled Jacobi coordinates can be transformed through
kinematic rotations:

$\displaystyle\left(\begin{array}[]{l}\mathbf{\xi_{2}}_{\tau+1}\\\
\mathbf{\xi_{1}}_{\tau+1}\end{array}\right)=\mathbf{T}\left(\begin{array}[]{l}\xi_{2\tau}\\\
\xi_{1\tau}\end{array}\right),$ (3)

where

$\mathbf{T}\left(\chi_{\tau+1,\tau}\right)=\left(\begin{array}[]{cc}\cos\left(\chi_{\tau+1,\tau}\right)\mathbf{1}&\sin\left(\chi_{\tau+1,\tau}\right)\mathbf{1}\\\
-\sin\left(\chi_{\tau+1,\tau}\right)\mathbf{1}&\cos\left(\chi_{\tau+1,\tau}\right)\mathbf{1}\end{array}\right)$
(4)

is a $6\times 6$ matrix and $\mathbf{1}$ is the $3\times 3$ unit matrix. The
kinematic angles $\chi_{\tau+1,\tau}$ are negative and obtuse, revealing the
mass of three particles:

$\begin{array}[]{l}\cos\chi_{\tau+1,\tau}=-\frac{\mu}{d_{\tau}d_{\tau+1}m_{\tau+2}},\\\
\sin\chi_{\tau+1,\tau}=-\frac{1}{d_{\tau}d_{\tau+1}},\end{array}$ (5)

where
$d_{\tau}=\left[\frac{m_{\tau}}{\mu}\left(1-\frac{m_{\tau}}{m_{\tau}+m_{\tau+1}+m_{\tau+2}}\right)\right]^{1/2}$
is a scaling factor.

Figure 1: Three sets of Jacobi coordinate vectors.

Delves hyperspherical coordinates can be defined in any set of mass-scaled
Jacobi coordinates. In this work, hyperspherical coordinates are defined in
the $A$-set ($\tau=A$) mass-scaled Jacobi coordinates, where two identical
atoms are connected through the Jacobi vector $\vec{\rho_{1}}$. We denote the
angle between $\vec{\rho}_{1}$ and $\vec{\rho}_{2}$ as $\theta$. The channel
functions are symmetric with respect to the $\theta$ direction in this
definition. After separation of the center of the mass motion, three of the
six coordinates are taken to be the Euler angles—$\alpha$, $\beta$, and
$\gamma$—that specify the orientation of the body-fixed frame relative to the
space-fixed frame. The remaining degrees of freedom can be represented by the
hyperradius $R$ and the two hyperangles $\theta$ and $\phi$, which are defined
as CDLIN1995

$\mu R=\mu_{1}\rho^{2}_{1}+\mu_{2}\rho^{2}_{2}$ (6)

and

$\tan\phi=\sqrt{\frac{\mu_{2}}{\mu_{1}}}\frac{\rho_{2}}{\rho_{1}},\;\;0\leq\phi\leq\frac{\pi}{2},$
(7)

respectively. $R$ is the only coordinate with the dimension of length, which
represents the size of the three-body system. Here, $\theta$, $\phi$ and the
three Euler angles $(\alpha,\beta,\gamma)$ can be collectively represented by
$\Omega$ $[\Omega\equiv(\theta,\phi,\alpha,\beta,\gamma)]$. In our method,
wave functions are expanded in the body frame $xyz$, where $\vec{\rho_{2}}$
lies along the $\vec{z}$-axis and the three particles lie on the $xz$ plane.

We introduce the reduced wave function
$\psi_{\upsilon^{\prime}}(R;\Omega)=\Psi_{\upsilon^{\prime}}(R;\Omega)R^{5/2}\sin\phi\cos\phi$,
and the Schrödinger equation is of the form:

$\bigg{[}-\frac{1}{2\mu}\frac{d^{2}}{dR^{2}}+\bigg{(}\frac{\Lambda^{2}-\frac{1}{4}}{2\mu
R^{2}}+V(R;\theta,\phi)\bigg{)}\bigg{]}\psi_{\upsilon^{\prime}}(R;\Omega)=E\psi_{\upsilon^{\prime}}(R;\Omega)\,,$
(8)

where $\Lambda^{2}$ is the squared “grand angular momentum operator”, whose
expression is given in Ref. CDLIN1995 . The three-body interaction
$V(R;\theta,\phi)$ in Eq.(8) is taken to be a sum of the three pairwise two-
body interactions.

Equation (8) is solved in the hyperspherical adiabatic representation. Similar
to the usual adiabatic approximation, the hyperspherical adiabatic potentials
$U_{\nu}(R)$ and channel functions $\Phi_{\nu}(R;\Omega)$ are defined as
solutions of the following adiabatic eigenvalue problem:

$\bigg{(}\frac{\Lambda^{2}-\frac{1}{4}}{2\mu
R^{2}}+V(R;\theta,\phi)\bigg{)}\Phi_{\nu}(R;\Omega)=U_{\nu}(R)\Phi_{\nu}(R;\Omega)\,.$
(9)

We define the normalized and symmetrized D-functions associated with our
choice of the body frame:

$\displaystyle\tilde{D}^{J\Pi}_{IM}(\alpha,\beta,\gamma)=\frac{1}{4\pi}\frac{\sqrt{2J+1}}{\left[1+(\sqrt{2}-1)\delta_{I0}\right]}\left[D^{J}_{IM}+(-1)^{I+J}\Pi
D^{J}_{-IM}\right],$ (10)

where $J$ is the total nuclear orbital angular momentum, $M$ is its projection
onto the laboratory-fixed axis, and $\Pi$ is the parity with respect to the
inversion of the nuclear coordinates. The quantum number $I$ denotes the
projection of $J$ onto the body-frame $z$ axis.

The channel functions are expanded in terms of D-functions as follows:

$\displaystyle\Phi_{\nu}^{J\Pi M}(R;\Omega)=\sum^{J}_{I=0}u_{\nu
I}(R;\theta,\phi)\tilde{D}^{J\Pi}_{IM}(\alpha,\beta,\gamma),$ (11)

and $u_{\nu I}(R;\theta,\phi)$ is expanded with B-spline functions,

$u_{\nu
I}(R;\theta,\phi)=\sum^{N_{\phi}}_{i}\sum^{N_{\theta}/2}_{j}c_{i,j}B_{i}(\phi)\left[B_{j}(\theta)+B_{N_{\theta}+1-j}\left(\pi-\theta\right)\right],$
(12)

where $N_{\theta}$ and $N_{\phi}$ are the sizes of the basis sets in the
$\theta$ direction and $\phi$ direction, respectively. The constructed
symmetric B-spline basis sets utilized in the $\theta$ direction reduce the
number of basis functions to $N_{\theta}/2$.

Following the method of Ref. PhysRevA.84.052721 , the $R$-matrix propagation
method combined with the SVD approach is used. We divide the hyperradius into
(N - 1) intervals with the set of grid points $R_{1}<R_{2}<\cdots<R_{N}$. In
the interval $[R_{i},R_{i+1}]$, the SVD method is used to solve Eq. (8). With
this solution, we can determine the R-matrix, which is defined as

$\displaystyle\underline{\mathcal{R}}(R)=\underline{\textsl{F}}(R)[\widetilde{\underline{\textsl{F}}}(R)]^{-1}\,,$
(13)

where matrices $\underline{\textsl{F}}$ and
$\widetilde{\underline{\textsl{F}}}$ can be calculated from the solution of
Eqs. (8) and (9) by

$\displaystyle F_{\nu,\upsilon^{\prime}}(R)=\int
d\Omega\Phi_{\nu}(R;\Omega)^{*}\psi_{\upsilon^{\prime}}(R;\Omega)\,,$ (14)
$\displaystyle\widetilde{F}_{\nu,\upsilon^{\prime}}(R)=\int
d\Omega\Phi_{\nu}(R;\Omega)^{*}\frac{\partial}{\partial
R}\psi_{\upsilon^{\prime}}(R;\Omega)\,.$ (15)

Over the interval $[R_{1},R_{2}]$, when the $\underline{\mathcal{R}}$ matrix
at $R_{1}$ is known, the $\underline{\mathcal{R}}$ matrix at another point
$R=R_{2}$ can be calculated as follows:

$\displaystyle\underline{\mathcal{R}}(R_{2})=\underline{\mathcal{R}}_{22}-\underline{\mathcal{R}}_{21}\left[\underline{\mathcal{R}}_{11}+\underline{\mathcal{R}}(R_{1})\right]^{-1}\underline{\mathcal{R}}_{12}.$
(16)

Using the recurrence relation (16) in the R-matrix propagation method, we can
obtain the $R$-matrix at the matching point $R_{m}$ where the wave function is
matched to the wave function in the asymptotic region, and the three-body
system is one dissociated atom plus a bound two-body system.

The asymptotic wave function of the atom + dimer scattering process in
$\tau$-set Jacobi coordinates can be written as

$\psi_{A}^{(\lambda)}(\rho_{1},\rho_{2})=\sum_{i=1}^{N}\frac{\varphi_{i}\left(\rho_{1}^{\tau}\right)\mathcal{Y}_{l_{1}^{\tau}l_{2}^{\tau}JM}\left(\hat{\Omega}_{1}^{\tau},\hat{\Omega}_{2}^{\tau}\right)\left[f_{\lambda}\left(k\rho_{2}^{\tau}\right)\delta_{i\lambda}-g_{i}\left(k\rho_{2}^{\tau}\right)K_{i\lambda}\right]}{\rho_{1}^{\tau}k\rho_{2}^{\tau}},$
(17)

where $\hat{\Omega}_{1}^{\tau}=\left(\theta_{1}^{\tau},\phi_{1}^{\tau}\right)$
and $\hat{\Omega}_{2}^{\tau}=\left(\theta_{2}^{\tau},\phi_{2}^{\tau}\right)$
are the orientation angles of the vectors $\mathbf{\rho_{1}}$ and
$\mathbf{\rho_{2}}$, respectively, for the $\tau$ arrangement.
$\varphi_{i}(\rho_{1}^{\tau})$ are the wave functions of the dimer, and $f$
and $g$ are the energy-normalized regular and irregular spherical Bessel
functions, respectively, and have the following form:

$f_{\lambda}\left(k\rho_{2}^{\tau}\right)=\sqrt{\frac{2\mu_{\tau,\tau+1\tau+2}}{k\pi}}k\rho_{2}^{\tau}j_{\ell_{\tau,\tau+1\tau+2}}(k\rho_{2}^{\tau}),$
(18)
$g_{i}\left(k\rho_{2}^{\tau}\right)=\sqrt{\frac{2\mu_{\tau,\tau+1\tau+2}}{k\pi}}k\rho_{2}^{\tau}n_{\ell_{\tau,\tau+1\tau+2}}(k\rho_{2}^{\tau}).$
(19)

The form of the angular part in the body frame is

$\displaystyle\mathcal{Y}_{l_{1}^{\tau}l_{2}^{\tau}JM}\left(\hat{\Omega}_{\mathrm{1}}^{\tau},\hat{\Omega}_{2}^{\tau}\right)$
$\displaystyle=\sqrt{\frac{16\pi^{2}}{2J+1}}\sum_{I}^{J}\frac{\tilde{D}_{IM}^{J\Pi}\left(\alpha,\beta,\gamma\right)}{1+(\sqrt{2}-1)\delta_{I0}}\mathcal{Y}_{l_{1}^{\tau}l_{2}^{\tau}JI}\left(\hat{\Omega}_{1}^{\tau(\mathrm{body})},\hat{\Omega}_{2}^{\tau(\mathrm{body})}\right)$
(20)
$\displaystyle=\sqrt{\frac{16\pi^{2}}{2J+1}}\sum_{I}^{J}\frac{\tilde{D}_{IM}^{J\Pi}\left(\alpha,\beta,\gamma\right)}{1+(\sqrt{2}-1)\delta_{I0}}\sum_{m}\left\langle
l_{1}^{\tau}\,m\,l_{2}^{\tau}\,I-m\mid J\,I\right\rangle$
$\displaystyle\;\;\;\;~{}\times
Y_{l_{1}^{\tau}m}\left(\theta_{1}^{\tau(\mathrm{body})},\phi_{1}^{\tau(\mathrm{body})}\right)Y_{l_{2}^{\tau}I-m}\left(\theta_{2}^{\tau(\mathrm{body})},\phi_{2}^{\tau(\mathrm{body})}\right),$

where the superscript (body) means these angles are measured in the body-fixed
frame.

For simplicity,
$S_{Il_{1}^{\tau}l_{2}^{\tau}JM}\left(\theta^{\tau},\phi^{\tau}\right)$ is
introduced:

$\displaystyle
S_{Il_{1}^{\tau}l_{2}^{\tau}JM}\left(\theta^{\tau},\phi^{\tau}\right)\equiv\sqrt{\frac{16\pi^{2}}{2J+1}}\frac{1}{1+(\sqrt{2}-1)\delta_{I0}}\sum_{m}\left\langle
l_{1}^{\tau}\,m\,l_{2}^{\tau}\,I-m\mid J\,I\right\rangle$ (21)
$\displaystyle\times
Y_{l_{1}^{\tau}m}\left(\theta_{1}^{\tau(\mathrm{body})},\phi_{1}^{\tau(\mathrm{body})}\right)Y_{l_{2}^{\tau}I-m}\left(\theta_{2}^{\tau(\mathrm{body})},\phi_{2}^{\tau(\mathrm{body})}\right).$

After transforming the asymptotic wave function into a body frame, the
matching process between the inner region wave function $\Psi=\sum_{\mu
I}F_{\mu I}\left(R_{m}\right)u_{\mu
I}\left(R_{m};\theta,\phi\right)\tilde{D}_{IM}^{J}\left(\alpha,\beta,\gamma\right)$
calculated in Delves coordinates and the asymptotic function $\psi_{A}$ in
Jacobi coordinates can be implemented:

$\frac{1}{R_{m}^{5/2}\sin\phi\cos\phi}\sum_{\sigma=1}^{N}H_{\sigma}^{\lambda}\Psi^{(\sigma)}\left(R_{m},\Omega\right)=\left.\psi_{A}^{(\lambda)}\left(\rho_{1},\rho_{2}\right)\right|_{R=R_{m}},$
(22)

where $H_{\sigma}^{\lambda}$ denotes the expansion coefficients; that is,

$\displaystyle\frac{\sum_{\mu I}\sum_{\sigma=1}^{N}H_{\sigma}^{\lambda}F_{\mu
I}^{(\sigma)}\left(R_{m}\right)u_{\mu
I}\left(R_{m};\theta,\phi\right)\tilde{D}_{IM}^{J\Pi}\left(\alpha,\beta,\gamma\right)}{R_{m}^{5/2}\sin\phi\cos\phi}$
(23)
$\displaystyle=\sum_{i=1}^{N}\varphi_{i}\left(\rho_{1}^{\tau}\right)\left[f_{\lambda}\left(k\rho_{2}^{\tau}\right)\delta_{i\lambda}-g_{i}\left(k\rho_{2}^{\tau}\right)K_{i\lambda}\right]\frac{\sum_{I}\tilde{D}_{IM}^{J\Pi}\left(\alpha,\beta,\gamma\right)S_{Il_{1}^{\tau}l_{2}^{\tau}JM}\left(\theta^{\tau},\phi^{\tau}\right)}{kR_{m}^{2}\cos\phi^{\tau}\sin\phi^{\tau}(\mu^{2}/\mu_{1}^{\tau}\mu_{2}^{\tau})^{1/2}}.$

Using the orthogonality and normalization of $u_{\mu
I}\left(R_{m};\theta,\phi\right)\tilde{D}_{IM}^{J\Pi}\left(\alpha,\beta,\gamma\right)$,
we can obtain the following relation:

$\displaystyle\sum_{\sigma=1}^{N}H_{\sigma}^{\lambda}F_{\mu
I}^{(\sigma)}\left(R_{m}\right)$
$\displaystyle=\left.R_{m}^{5/2}\int(\sin\phi\cos\phi)u_{\mu
I}\left(R_{m};\theta,\phi\right)\tilde{D}_{IM}^{J\Pi}\psi_{A}^{(\lambda)}\left(\rho_{1},\rho_{2}\right)\right|_{R=R_{m}}\mathrm{d}\phi\mathrm{d}\hat{\Omega}$
(24) $\displaystyle\equiv R_{m}^{1/2}\left(J_{\mu
I}^{\lambda}-\sum_{i=1}^{N}N_{\mu I}^{i}K_{i\lambda}\right),$

where

$\displaystyle J_{\mu I}^{\lambda}=\int$
$\displaystyle\frac{\sin\phi\cos\phi}{k\sin\phi^{\tau}\cos\phi^{\tau}(\mu^{2}/\mu_{1}^{\tau}\mu_{2}^{\tau})^{1/2}}u_{\mu
I}\left(R_{m};\theta,\phi\right)$ (25)
$\displaystyle\times\varphi_{\lambda}\left(\rho_{1}^{\tau}\right)f_{\lambda}\left(k\rho_{2}^{\tau}\right)S_{Il_{1}^{\tau}l_{2}^{\tau}JM}\left(\theta^{\tau},\phi^{\tau}\right)\sin\theta\mathrm{d}\theta\mathrm{d}\phi,$
$\displaystyle N_{\mu I}^{i}=\int$
$\displaystyle\frac{\sin\phi\cos\phi}{k\sin\phi^{\tau}\cos\phi^{\tau}(\mu^{2}/\mu_{1}^{\tau}\mu_{2}^{\tau})^{1/2}}u_{\mu
I}\left(R_{m};\theta,\phi\right)$ (26)
$\displaystyle\times\varphi_{i}\left(\rho_{1}^{\tau}\right)g_{i}\left(k\rho_{2}^{\tau}\right)S_{Il_{1}^{\tau}l_{2}^{\tau}JM}\left(\theta^{\tau},\phi^{\tau}\right)\sin\theta\mathrm{d}\theta\mathrm{d}\phi.$

At the matching point $R_{m}$, the logarithmic derivative of the inner and
outer region wave functions should be equal; therefore, the derivative of the
asymptotic wave function is also needed:

$\displaystyle J_{\mu I}^{\lambda\prime}=$
$\displaystyle\int\frac{\sin\phi\cos\phi u_{\mu
I}\left(R_{m};\theta,\phi\right)}{k\sin\phi^{\tau}\cos\phi^{\tau}}\left(\sqrt{\frac{\mu_{2}^{\tau}}{\mu}}\cos\phi^{\tau}\varphi_{\lambda}^{\prime}\left(\rho_{1}^{\tau}\right)f_{\lambda}\left(k\rho_{2}^{\tau}\right)\right.$
(27)
$\displaystyle\left.+\sqrt{\frac{\mu_{1}^{\tau}}{\mu}}\sin\phi^{\tau}\varphi_{\lambda}\left(\rho_{1}^{\tau}\right)f_{\lambda}^{\prime}\left(k\rho_{2}^{\tau}\right)\right)_{R=R_{m}}S_{Il_{1}^{\tau}l_{2}^{\tau}JM}\left(\theta^{\tau},\phi^{\tau}\right)\sin\theta\mathrm{d}\theta\mathrm{d}\phi+\frac{J_{\mu
I}^{\lambda}}{2R_{m}}$

and

$\displaystyle N_{\mu I}^{i\prime}=\int$ $\displaystyle\frac{\sin\phi\cos\phi
u_{\mu
I}\left(R_{m};\theta,\phi\right)}{k\sin\phi^{\tau}\cos\phi^{\tau}}\left(\sqrt{\frac{\mu_{2}^{\tau}}{\mu}}\cos\phi^{\tau}\varphi_{i}^{\prime}\left(\rho_{1}^{\tau}\right)g_{i}\left(k\rho_{2}^{\tau}\right)\right.$
(28)
$\displaystyle\left.+\sqrt{\frac{\mu_{1}^{\tau}}{\mu}}\sin\phi^{\tau}\varphi_{i}\left(\rho_{1}^{\tau}\right)g_{i}^{\prime}\left(k\rho_{2}^{\tau}\right)\right)_{R=R_{m}}S_{Il_{1}^{\tau}l_{2}^{\tau}JM}\left(\theta^{\tau},\phi^{\tau}\right)\sin\theta\mathrm{d}\theta\mathrm{d}\phi+\frac{N_{\mu
I}^{i}}{2R_{m}}.$

The matrix form of Eq. (24) is

$\textbf{F}\textbf{H}=R_{m}^{1/2}[\textbf{J}-\textbf{N}\textbf{K}].$ (29)

The derivative with respect to $R$ at the matching point $R_{m}$ is

$\textbf{F}^{\prime}\textbf{H}=R_{m}^{1/2}[\textbf{J}{{}^{\prime}}-\textbf{N}{{}^{\prime}}\textbf{K}].$
(30)

According the definition of the R matrix,
$\textbf{R}=\textbf{F}\textbf{F}^{\prime-1}$, with (29) and (30), we can
obtain the reaction matrix,

$\textbf{K}=\frac{\textbf{J}-\textbf{RJ}^{\prime}}{\textbf{N}-\textbf{RN}^{\prime}},$
(31)

and the scattering matrix,

$\displaystyle\textbf{S}=\frac{\mathbf{1}+i\textbf{K}}{\mathbf{1}-i\textbf{K}}\,,$
(32)

where $\mathbf{1}$ is the $3\times 3$ unit matrix. The relation between the
atom + dimer scattering phase shift $\delta_{0}$ and the diagonal element
scattering matrix S is

$S_{0\leftarrow 0}^{0+}=\exp\left(2i\delta_{0}\right).$ (33)

With the phase shift $\delta_{0}$, we can obtain the atom + dimer scattering
length $a_{ad}$ through

$a_{ad}=-\lim_{k_{\tau,\tau+1\tau+2}\rightarrow
0}\frac{\tan\delta_{0}}{k_{\tau,\tau+1\tau+2}},$ (34)

and the total cross section $\sigma_{2}$ is

$\sigma_{2}=\sum_{J,\Pi}\sigma_{2}^{J\Pi}=\sum_{J,\Pi}\frac{(2J+1)\pi}{k_{\tau,\tau+1\tau+2}^{2}}\left|S_{0\leftarrow
0}^{J\Pi}-1\right|^{2}.$ (35)

##  3 Results and discussion

###  3.1 Pair potentials

For the helium dimer potential $v_{\textsl{HeHe}}(r)$, we use the CCSAPT
potential of Jeziorska et al. doi:10.1063/1.2770721 . The interaction between
He and the spin-polarized tritium (T$\uparrow$) is identical to that between H
and He. We choose the H–He potential developed by Cvetko et al.
doi:10.1063/1.466505 . The H and Li atoms are assumed to be spin-stretched.
Their short-range potentials are determined from ab initio calculations
doi:10.1063/1.1388044 ; doi:10.1063/1.1697142 , and their long-range behavior
is determined by the usual dispersion potentials PhysRevA.71.032709 ;
doi:10.1063/1.1388044 . All pairwise interaction potentials used in this work
are shown in Fig. 2. Their bound state energies $E_{\upsilon l}=E_{00}$ and
scattering length $a$ calculated with potentials are summarized in Table 1.

Figure 2: Pairwise interaction potentials for He–He, He–T$\uparrow$, H–H, H–Li Table 1: Two-body bound state energies E00 (in a.u.) and scattering length $a$ (in a.u.) for He–He, He–T$\uparrow$, H–H, and H–Li interactions. system | E00 | $a$
---|---|---
4He2 | $-5.47114\times 10^{-9}$ | 165.5
4He3He | ——— | -34
4He3T$\uparrow$ | ——— | -25
H$\uparrow$-H$\uparrow$ | ——— | 1.6
Li-H$\uparrow$ | $-1.274\times 10^{-7}$ | 63.5

###  3.2 Matching in $A$-set: 3He4He2 and T$\uparrow^{4}$He2 system

In our definition of the hyperspherical coordinates, the two identical atoms
are connected with Jacobi vector $\vec{\rho_{1}}$. Thus, the inner region wave
functions are matched with the asymptotic solutions in $A$-set Jacobi
coordinates for 3He4He2 and T$\uparrow^{4}$He2 systems. Several papers have
reported on the process of 3He atom scattering from the 4He2 dimer, which is a
good example to test our procedure. Kolganova and Sandhans et al. 2004Binding
; 2006Scattering ; Kolganova2009Ultracold  calculated the scattering phase
shifts and scattering length for 3He + 4He2 using the two-dimensional partial-
wave integral-differential Faddeev equations based on the SAPT2 and LM2M2
potentials. They estimated the scattering length to be between 35.9 a.u. and
37 a.u. based on these two kinds of potentials. Soon thereafter, Suno
PhysRevA.78.062701  calculated the scattering length by solving the coupled-
channel hyperradial equations using a combination of the finite element method
Burke1999Theoretical  and the R-matrix method RevModPhys.68.1015 . They used
the improved He-He potential doi:10.1063/1.2770721  and predicted a 3He + 4He2
scattering length value of 40 a.u..

Due to the similarities between 4He and T$\uparrow$ atoms, similar behaviors
of mixed T$\uparrow$4He2 and 3He4He2 clusters are expected. For example, both
systems have been found to possess one weakly bound state and exhibit a larger
spatial extension with universal halo properties Hiroya2014A ;
PhysRevLett.113.253401 ; doi:10.1063/1.3530837 . However, compared with the
well-studied 3He + 4He2 elastic scattering process, no scattering observables
are available for the T$\uparrow$ atom scattering from the 4He2 dimer.

The potential curves of T$\uparrow$4He2 and 3He4He2 are presented in Fig. 3.
The lowest potential curves correspond asymptotically to the atom–dimer
channel for T$\uparrow$-4He2 and 3He-4He2, and the other potential curves
represent three-body continuum states. From Fig. 3, the potential well of
T$\uparrow$4He2 is shallower than that of the 3He4He2 system.

Figure 3: Adiabatic hyperspherical potential curves $U_{\nu}(R)$ for (a)
3He4He2, $J^{\Pi}=0^{+}$ and (b) T$\uparrow$4He2, $J^{\Pi}=0^{+}$. The values
indicate the asymptotic behavior of the potential curves, as given in Eq.
(17).

For ultracold atom–dimer collisions, the convergence of scattering observables
depends critically on the accuracy of the adiabatic potentials. Thus, accurate
potential curves and channel functions are highly desirable. According to the
behavior of the channel function for these weakly bound systems, different
B-spline knot distributions are used at short- and long-range hyperradii. For
small hyperradii $R$, uniform knots are distributed; for large hyperradii $R$,
the knot distribution is designed so that it becomes dense around the two-body
coalescence points where the channel function is localized. Table 2 shows the
convergence of lowest hyperspherical potential curves as functions of basis
sets for T$\uparrow$4He2 and 3He4He2 systems. The basis sets $N_{\theta}=168$
and $N_{\phi}=504$ are chosen as the final calculation, and the potential
curves have at least six significant digits. The convergence of the scattering
observables with respect to the number of adiabatic channels and sectors is
also tested. We typically use 13 channels and 230 sectors distributed as
$R_{i}\propto i^{3}$ from $R=2$ a.u. to $R=500$ a.u..

Table 2: Convergence test of adiabatic potentials (in a.u.) of T$\uparrow$4He2
and 3He4He2 as functions of the B-spline basis size ($N_{\theta}$,
$N_{\phi}$). [a] abbreviates $\times 10^{a}$.  T$\uparrow$4He2
---
B($\theta$,$\phi$) | R=20 | R=100 | R=200 | R=300 | R=500
(106, 304) | -1.061490[-6] | -1.212912[-8] | -6.770491[-9] | -6.030653[-9] | -5.668053[-9]
(106, 504) | -1.061490[-6] | -1.212912[-8] | -6.770492[-9] | -6.030666[-9] | -5.668068[-9]
(186, 504) | -1.061490[-6] | -1.212912[-8] | -6.770492[-9] | -6.030666[-9] | -5.668068[-9]
3He4He2
$B(\theta,\phi)$ | R=20 | R=100 | R=200 | R=300 | R=500
(106, 304) | -9.431873[-7] | -1.222582[-8] | -6.782697[-9] | -6.032252[-9] | -5.668068[-9]
(106,404) | -9.431873[-7] | -1.222582[-8] | -6.782720[-9] | -6.032406[-9] | -5.668099[-9]
(106, 504) | -9.431873[-7] | -1.222582[-8] | -6.782721[-9] | -6.032418[-9] | -5.668105[-9]
(186, 504) | -9.431873[-7] | -1.222582[-8] | -6.782722[-9] | -6.032419[-9] | -5.668106[-9]

Figure 4 represents the $J=0$ cross sections for elastic 3He + 4He2 and
T$\uparrow$ \+ 4He2 scattering as functions of the collision energy (E-E00).
In the ultracold limit, $\sigma^{0+}$ obeys the threshold behavior as
$\sigma^{0+}\propto(E-E_{00})^{0}$.

Figure 4: $J=0$ cross sections for elastic 3He + 4He2 and T$\uparrow$ \+ 4He2
scattering as functions of the collision energy (E-E00).

Table 3 shows the convergence of scattering lengths
$a_{{}^{3}\textsl{He}+^{4}\textsl{He}_{2}}$ and
$a_{\textsl{T}\uparrow+^{4}\textsl{He}_{2}}$ as a function of the matching
distance. The scattering length converges at $R_{m}=500\;a.u.$ for both
systems. As shown in Refs. PhysRevA.78.062701 ; PhysRevA.83.032703 , the
numerical solutions of these kinds of systems are usually matched to the
asymptotic analytical solutions at $R_{m}=5000\sim 10000$ a.u. in the
hyperspherical coordinates boundary condition.

A comparison of our calculations with the results available in the literature
is given in Table 4. For 3He + 4He2 elastic scattering, Suno et al.
PhysRevA.78.062701  obtained
$a_{{}^{3}\textsl{He}+^{4}\textsl{He}_{2}}=40\;a.u.\;$ using the potential
from Ref. doi:10.1063/1.2770721 . With the same potential, the scattering
length we calculated is
$a_{{}^{3}\textsl{He}+^{4}\textsl{He}_{2}}=34.6\;a.u.$. Sandhas et al.
Kolganova2009Ultracold  obtained
$a_{{}^{3}\textsl{He}+^{4}\textsl{He}_{2}}=37\;a.u.$ and
$a_{{}^{3}\textsl{He}+^{4}\textsl{He}_{2}}=35.9\;a.u.$ using LM2M2 and SAPT2
potentials, respectively.

For 3T$\uparrow$ \+ 4He2 scattering, we obtain a scattering length value of
$a_{\textsl{T}\uparrow+^{4}\textsl{He}_{2}}=166\;a.u.$, which is larger than
that of 3He + 4He2 scattering. This result supports Suno’s result that the
3T$\uparrow$ \+ 4He2 bound state extends to larger distances than the 3He +
4He2 bound state.

Table 3: Convergence test of the scattering lengths (in a.u.) of $a_{\textsl{T}\uparrow+^{4}\textsl{He}_{2}}$ and $a_{{}^{3}\textsl{He}+^{4}\textsl{He}_{2}}$ as functions of matching distances $R_{m}$(in a.u.). $R_{m}$ | $a_{{}^{3}\textsl{He}+^{4}\textsl{He}_{2}}$ | $a_{\textsl{T}\uparrow+^{4}\textsl{He}_{2}}$
---|---|---
200 | 40.9 |
300 | 35.2 | 168.3
400 | 34.1 | 166.6
500 | 34.6 | 166.8
Table 4: Atom–molecule zero-energy scattering lengths (in a.u.) of $a_{\textsl{T}\uparrow+^{4}\textsl{He}_{2}}$ and $a_{{}^{3}\textsl{He}+^{4}\textsl{He}_{2}}$ . Other calculations are also shown in the table for comparison. | He–He potential | $a_{{}^{3}\textsl{He}+^{4}\textsl{He}_{2}}$ | $a_{\textsl{T}\uparrow+^{4}\textsl{He}_{2}}$
---|---|---|---
present | CCSAPT doi:10.1063/1.2770721  | 34 | 166
Ref. PhysRevA.78.062701  | CCSAPT doi:10.1063/1.2770721  | 40 |
Ref. Kolganova2009Ultracold  | LM2M2 doi:10.1063/1.460139  | 37 |
Ref. Kolganova2009Ultracold  | SAPT2 doi:10.1063/1.474444  | 35.9 |

###  3.3 Matching in the $B$-set: LiHH system

In the Delves hyperspherical coordinates defined in $A$-set Jacobi
coordinates, where two identical atoms are connected with $\vec{\rho_{1}}$,
the asymptotic wave functions of the scattering process A + AC $\rightarrow$ A
+ AC (the A and C atoms are bounded) involve the transformation between the
$A$-set and $B$-set. The spin-stretched case of H atom scattering from LiH is
such an example where the lowest adiabatic potentials asymptotically depend on
the binding energies of the H–Li two-body bound states. Thus, the asymptotic
wave function of the dissociated system is better represented in the $B$-set
as follows:

$\psi_{A}^{(\lambda)}(\rho_{1}^{B},\rho_{2}^{B})=\sum_{i=1}^{N}\frac{\varphi_{i}\left(\rho_{1}^{B}\right)\mathcal{Y}_{l_{1}^{B}l_{2}^{B}JM}\left(\hat{\Omega}_{1}^{B},\hat{\Omega}_{2}^{B}\right)\left[f_{\lambda}\left(k\rho_{2}^{B}\right)\delta_{i\lambda}-g_{i}\left(k\rho_{2}^{B}\right)K_{i\lambda}\right]}{\rho_{1}^{B}k\rho_{2}^{B}}.$
(36)

The transformations between the $A$-set and $B$-set Jacobi coordinates can be
implemented by the kinematic rotations given in Eq. (4). The components of
$\mathbf{\xi_{2}}$ and $\mathbf{\xi_{1}}$ in the $A$-set body frame can be
written as

$\left(\begin{array}[]{l}\mathbf{\xi_{2}}_{A}\\\
\mathbf{\xi_{1}}_{A}\end{array}\right)=\left(\begin{array}[]{l}\xi_{2Ax}=\xi_{2A}\sin\theta\\\
\xi_{2Ay}=0\\\ \xi_{2Az}=\xi_{2A}\cos\theta\\\ \xi_{1Ax}=0\\\ \xi_{1Ay}=0\\\
\xi_{1Az}=\xi_{1A}\end{array}\right).$ (37)

With Eqs. (3) and (4), we can obtain the components of $\mathbf{\xi_{2}}$ and
$\mathbf{\xi_{1}}$ in the $B$-set body frame as follows:

$\displaystyle\left(\begin{array}[]{l}\mathbf{\xi_{2}}_{B}\\\
\mathbf{\xi_{1}}_{B}\end{array}\right)=\left(\begin{array}[]{cc}\cos\left(\chi_{B,A}\right)\mathbf{1}&\sin\left(\chi_{B,A}\right)\mathbf{1}\\\
-\sin\left(\chi_{B,A}\right)\mathbf{1}&\cos\left(\chi_{B,A}\right)\mathbf{1}\end{array}\right)\left(\begin{array}[]{l}\xi_{2Ax}=\xi_{2A}\sin\theta\\\
\xi_{2Ay}=0\\\ \xi_{2Az}=\xi_{2A}\cos\theta\\\ \xi_{1Ax}=0\\\ \xi_{1Ay}=0\\\
\xi_{1Az}=\xi_{1A}\end{array}\right)$ (38)

and

$\displaystyle\left(\begin{array}[]{l}\xi_{2Bx}=\cos\left(\chi_{B,A}\right)\xi_{2A}\sin\theta\\\
\xi_{2By}=0\\\
\xi_{2Bz}=\cos\left(\chi_{B,A}\right)\xi_{2A}\cos\theta+\sin\left(\chi_{B,A}\right)\xi_{1A}\\\
\xi_{1Bx}=-\sin\left(\chi_{B,A}\right)\xi_{2A}\sin\theta\\\ \xi_{1By}=0\\\
\xi_{1Bz}=-\sin\left(\chi_{B,A}\right)\xi_{2A}\cos\theta+\cos\left(\chi_{B,A}\right)\xi_{1A}\end{array}\right).$
(39)

With these equations, the expression of $\rho_{1B}$, $\rho_{2B}$, $J_{\mu
I}^{\lambda}$, $N_{\mu I}^{i}$, $J_{\mu I}^{\lambda\prime}$ and $N_{\mu
I}^{i\prime}$ in $B$-set Jacobi coordinates can be obtained.

The adiabatic hyperspherical potentials $U_{\nu}(R)$ of the H–H–Li system are
presented in Fig. 5. The basis sets $N_{\theta}=168$ and $N_{\phi}=504$ are
used, giving the potential curve at least six significant digits.

Figure 5: Adiabatic hyperpotential curves $U(R)$ for the H–H–Li system with
$J^{\Pi}=0^{+}$.

We plot the $J^{\Pi}=0^{+}$ partial wave cross sections for elastic collisions
between H and LiH in Fig. 6. For this system, 18 channels and 230 sectors are
used to ensure that the scattering length has at least two digits. Table 5
shows the convergence test of the atom–molecule scattering length
$a_{\textsl{H+LiH}}$ as a function of the matching distance $R_{m}$. The
scattering length is converged at the matching distance $R_{m}=500$ a.u. Note
that Yujun et al. PhysRevA.83.032703  calculated the elastic cross sections
for H + LiH collisions in hyperspherical coordinate conditions. They matched
the numerical solutions to the asymptotic analytical solutions at
$R_{m}=5\times 10^{3}$ a.u., and given the H + LiH scattering length of $80$
a.u. with the same $v_{HH}(r)$ and $v_{LiH}(r)$ potentials, the present result
shows good agreement with their results.

Figure 6: Cross section $\sigma_{2}$ for elastic H + LiH scattering as a function of the collision energy $(E-E_{00})$ for $J^{\Pi}=0^{+}$.  Table 5: Convergence of the $J^{\Pi}=0^{+}$ scattering length between H and LiH $a_{\textsl{H+LiH}}$ (in a.u.) with respect to the matching distance $R_{m}$ (in a.u.). $R_{m}$ | $300$ | $400$ | $500$
---|---|---|---
$a_{\textsl{H+LiH}}$ | $79.7$ | $80.9$ | $80.3$

##  4 Conclusions

In this work, we present an efficient method for solving the coupled-channel
Schrödinger equation for atom–molecule elastic collisions. We use Delves
hyperspherical coordinates, expand the wave function in a coupled-channel
basis and propagate the coupled-channel equations with the R-matrix
propagation technique. To avoid derivative coupling terms, we adopt the smooth
variable discretization method, which discretizes the propagation variable
before expanding in the basis. In the matching procedure, the asymptotic wave
functions are expressed in the rotated Jacobi coordinates. Test calculations
of elastic atom–molecule collisions are performed. For 3He(T$\uparrow$) atom
scattering from 4He2, the asymptotic wave functions are also expressed in the
$A$-set Jacobi coordinates. No coordinate rotation is needed in this case. For
the spin-stretched case of H atom scattering from LiH, the asymptotic wave
functions must be expressed in the $B$-set Jacobi coordinates to describe the
final scattering shape. Coordinate rotation between the $A$-set and $B$-set is
needed for this type of scattering process. The convergence of the scattering
length as a function of the propagation distance is studied. We find that the
method is reliable and can improve the convergence as a function of matching
distance. We compare our results with those of other calculations. The
scattering length of H–HLi shows good agreement with that of hyperspherical
coordinate boundary conditions with less computational expense. The scattering
observables of T$\uparrow$-4He2 are scarce, whose scattering length and cross
section values are given for the first time.

###### Acknowledgements.

We thank C. H. Greene for helpful discussions. Hui-Li Han was supported by the
National Natural Science Foundation of China under Grant No. 11874391 and the
National Key Research and Development Program of China under Grant No.
2016YFA0301503. Ting-Yun Shi was supported by the Strategic Priority Research
Program of the Chinese Academy of Sciences under Grant No. XDB21030300.

## References

  * (1) H. Suno, B.D. Esry, C.H. Greene, Phys. Rev. Lett. 90, 053202 (2003)
  * (2) J.P. D’Incao, B.D. Esry, Phys. Rev. Lett. 103, 083202 (2009)
  * (3) C.H. Greene, P. Giannakeas, J. Pérez-Ríos, Rev. Mod. Phys. 89, 035006 (2017)
  * (4) P. Giannakeas, C.H. Greene, Few Body Sys. 58(1) (2017)
  * (5) J. Voigtsberger, S. Zeller, J. Becht, N. Neumann, F. Sturm, H.K. Kim, M. Waitz, F. Trinter, M. Kunitski, A. Kalinin, Nat. Commun. 5, 5765 (2014)
  * (6) Y. Li, H. Song, Q. Gou, H. Han, T. Shi, Phys. Rev. A 79, 024501 (2009)
  * (7) M.S. Wu, H.L. Han, C.B. Li, T.Y. Shi, Phys. Rev. A 90, 062506 (2014)
  * (8) M. Kunitski, S. Zeller, J. Voigtsberger, A. Kalinin, L.P.H. Schmidt, M. Schöffler, A. Czasch, W. Schöllkopf, R.E. Grisenti, T. Jahnke, D. Blume, R. Dörner, Science 348(6234), 551 (2015)
  * (9) H. Suno, B.D. Esry, Phys. Rev. A 78, 062701 (2008)
  * (10) Y. Wang, J.P. D’Incao, B.D. Esry, Phys. Rev. A 83, 032703 (2011)
  * (11) W. Sandhas, E.A. Kolganova, Y.K. Ho, A.K. Motovilov, Few Body Sys. 34(1/3), 137 (2004)
  * (12) E.A. Kolganova, A.K. Motovilov, W. Sandhas, Few Body Sys. 38(2/4), 205 (2006)
  * (13) E.A. Kolganova, A.K. Motovilov, W. Sandhas, Phys. Part. Nucl. 40, 206 (2009)
  * (14) M.A. Shalchi, M.T. Yamashita, T. Frederico, L. Tomio, Phys. Rev. A 102, 062814 (2020)
  * (15) P. Stipanović, L.V. Markić, I. Bešlić, J. Boronat, Phys. Rev. Lett. 113, 253401 (2014)
  * (16) H. Suno, Few Body Sys. 55(4), 229 (2014)
  * (17) P. Stipanović, L.V. Markić, J. Boronat, B. Kez̆ić, J. Chem. Phys. 134(5), 054509 (2011)
  * (18) C.D. Lin, Phys. Rep. 257(1), 1 (1995)
  * (19) H. Suno, J. Chem. Phys. 132(22), 224311 (2010)
  * (20) Y. Li, D. Huang, Q. Gou, H. Han, T. Shi, Phys. Rev. A 84, 014501 (2011)
  * (21) P. Barletta, C. Romero-Redondo, A. Kievsky, M. Viviani, E. Garrido, Phys. Rev. Lett. 103, 090402 (2009)
  * (22) C. Romero-Redondo, E. Garrido, P. Barletta, A. Kievsky, M. Viviani, Phys. Rev. A 83, 022705 (2011)
  * (23) P. Barletta, A. Kievsky, Few-Body Syst. 45(1), 25 (2009)
  * (24) O.I. Tolstikhin, N. Hiroki, J. Chem. Phys. 108(21), 8899 (1998)
  * (25) G.D. Billing, Chem. Phys. Lett. 75(2), 254 (1980)
  * (26) D. Bondi, J. Connor, Chem. Phys. Lett. 92(6), 570 (1982)
  * (27) C.L. Shoemaker, N. AbuSalbi, D.J. Kouri, J. Phys. Chem. 87(26), 5389 (1983)
  * (28) F.D. Colavecchia, F. Mrugala, G.A. Parker, R.T. Pack, J. Chem. Phys. 118(23), 10387 (2003)
  * (29) G.A. Parker, R.B. Walker, B.K. Kendrick, R. T Pack, J. Chem. Phys. 117(13), 6083 (2002)
  * (30) Y. Zhou, C.D. Lin, J.Phys. B 27(20), 5065 (1994)
  * (31) Y. Zhou, C.D. Lin, J. Phys. B 28(22), 4907 (1995)
  * (32) A. Igarashi, N. Toshima, Phys. Rev. A 50, 232 (1994)
  * (33) B. Zhou, C.D. Lin, J.Z. Tang, S. Watanabe, M. Matsuzawa, J. Phys. B 26(16), 2555 (1993)
  * (34) B. Zhou, C.D. Lin, J. Phys. B 26(16), 2575 (1993)
  * (35) Z.X. Zhao, A. Igarashi, C.D. Lin, Phys. Rev. A 62, 042706 (2000)
  * (36) Y. Zhou, S. Watanabe, O.I. Tolstikhin, T. Morishita, Phys. Rev. A 92, 032713 (2015)
  * (37) O.I. Tolstikhin, S. Watanabe, M. Matsuzawa, J. Phys. B 29(11), L389 (1996)
  * (38) J.C. Light, R.B. Walker, J. Chem. Phys. 65(10), 4272 (1976)
  * (39) J. Wang, J.P. D’Incao, C.H. Greene, Phys. Rev. A 84, 052721 (2011)
  * (40) M. Jeziorska, W. Cencek, K. Patkowski, B. Jeziorski, K. Szalewicz, J. Chem. Phys. 127(12), 124303 (2007)
  * (41) D. Cvetko, A. Lausi, A. Morgante, F. Tommasini, P. Cortona, M.G. Dondi, J. Chem. Phys. 100(3), 2052 (1994)
  * (42) N. Geum, G.H. Jeung, A. Derevianko, R. Côté, A. Dalgarno, J. Chem. Phys. 115(13), 5984 (2001)
  * (43) W. Kolos, L. Wolniewicz, J. Chem. Phys. 43(7), 2429 (1965)
  * (44) J. Mitroy, M.W.J. Bromley, Phys. Rev. A 71, 032709 (2005)
  * (45) James P. Burke, Jr, Theoretical investigation of cold alkali atom collisions.  Ph.D. thesis, University of Colorado (1999)
  * (46) M. Aymar, C.H. Greene, E. Luc-Koenig, Rev. Mod. Phys. 68, 1015 (1996)
  * (47) R.A. Aziz, M.J. Slaman, J. Chem. Phys. 94(12), 8047 (1991)
  * (48) A.R. Janzen, R.A. Aziz, J. Chem. Phys. 107(3), 914 (1997)