00footnotetext: Junyao Pan. E-mail addresses<EMAIL_ADDRESS>
# The full automorphism groups of general position graphs
Dedicated to my father Hongqi Pan’s 75th birthday††thanks: The research of the
work was partially supported by the Hainan Provincial Natural Science
Foundation of China (No. 122RC652) and the National Natural Science Foundation
of China (No. 12061030; No. 61962018).

Junyao Pan

School of Sciences, University of Wuxi, Wuxi, Jiangsu,
214105 People’s Republic of China

Abstract: Let $S$ be a non-empty finite set. A flag of $S$ is a set $f$ of
non-empty proper subsets of $S$ such that $X\subseteq Y$ or $Y\subseteq X$ for
all $X,Y\in f$. The set $\\{|X|:X\in f\\}$ is called the type of $f$. Two
flags $f$ and $f^{\prime}$ are in general position with respect to $S$ if
$X\cap Y=\emptyset$ or $X\cup Y=S$ for all $X\in f$ and $Y\in f^{\prime}$. For
a fixed type $T$, Klaus Metsch defined the general position graph
$\Gamma(S,T)$ whose vertices are the flags of $S$ of type $T$ with two
vertices being adjacent when the corresponding flags are in general position.
In this paper, we characterize the full automorphism groups of $\Gamma(S,T)$
in the case that $|T|=2$. In particular, we solve an open problem proposed by
Klaus Metsch.

Keywords: Flag; General Position Graph; Automorphism Group.

Mathematics Subject Classification: 20B25, 05D05.

##  1 Introduction

Throughout this paper, $[n]=\\{1,2,...,n\\}$ denotes the standard $n$-element
set and $\overline{A}$ stands for the complement of $A$ in $[n]$ where
$A\subseteq[n]$. Moreover, $C^{k}_{m}=\frac{m(m-1)\cdot\cdot\cdot(m-k+1)}{k!}$
expresses the binomial coefficient.

For two positive integers $n$ and $k$ with $n\geq 2k$, the _Kneser graph_
$KG(n,k)$ has as vertices the $k$-subsets of $[n]$ with edges defined by
disjoint pairs of $k$-subsets. It is well-known that the problem of the
independence number of $KG(n,k)$ reduces to the famous Erdős-Ko-Rado Theorem.
From this perspective, Klaus Metsch [8] introduced the _general position
graph_ of flags of $[n]$ to generalize the famous Erdős-Ko-Rado Theorem. A
_flag_ of $[n]$ is a set $f$ of non-empty proper subsets of $[n]$ such that
$A\subseteq B$ or $B\subseteq A$ for all $A,B\in f$. The set $\\{|A|:A\in
f\\}$ is called the type of $f$. Two flags $f$ and $f^{\prime}$ are in
_general position_ with respect to $[n]$ if $A\cap B=\emptyset$ or $A\cup
B=[n]$ for all $A\in f$ and $B\in f^{\prime}$. For a fixed type
$T\subseteq[n-1]$ the _general position graph_ whose vertices are the flags of
$[n]$ of type $T$ with two vertices being adjacent when the corresponding
flags are in general position will be abbreviated as $\Gamma(n,T)$. If
$|T|=1$, then this graph is isomorphic to a corresponding Kneser graph. Klaus
Metsch [8] not only described the independence number of $\Gamma(n,T)$ in some
situations but also proposed several interesting open problems, such as the
following one:

######  Question 1.1.

([8, Problem 5] ) Is it true that the graphs $\Gamma(n,T)$ have $S_{n}$ as
automorphism group, where $T=\\{a,b\\}$ with $n\geq a+b+1$ and
$a<\frac{n}{2}<b$? Can [8, Remark 5.13] be used to show this?

Let $\Gamma=(V,E)$ be a undirected graph with vertex set $V$ and edge set $E$.
If there exists a bijection $\alpha$ from $V$ to $V$ such that
$(f^{\alpha},g^{\alpha})\in E$ if and only if $(f,g)\in E$ for all $f,g\in V$,
then $\alpha$ is called an automorphism of $\Gamma$. Let $Aut(\Gamma)$ denote
the full automorphism group of $\Gamma$. Actually, the research on the
automorphism groups of graphs has always been an interesting topic for many
scholars in group theory and graph theory, for examples [3, 4, 6, 7, 9, 11].
Thereby, we are interested in Question 1.1.

Review some notions and notations about permutation groups, for details see
[1, 2]. Let $G$ be a transitive permutation group acting on $[n]$. A non-empty
subset $\Delta$ of $[n]$ is called a _block_ for $G$ if for each $\alpha\in G$
either $\Delta^{\alpha}=\Delta$ or $\Delta^{\alpha}\cap\Delta=\emptyset$.
Clearly, the singletons $\\{i\\}$ $(i\in[n])$ and $[n]$ are blocks, and so
these blocks are called the _trivial_ blocks. Any other block is called
_nontrivial_. Put $\Sigma=\\{\Delta^{\alpha}:\alpha\in G\\}$ where $\Delta$ is
a block of $G$. We call $\Sigma$ the system of blocks containing $\Delta$.
Clearly, $G$ reduces a permutation group acting on $\Sigma$, denoted by
$G|_{\Sigma}$. In addition, there exists a natural homomorphism from $G$ to
$G|_{\Sigma}$, and the kernel of this homomorphism consists of all
permutations in $G$ which fix every block in $\Sigma$. In this note, we divide
three cases to construct systems of blocks of $Aut(\Gamma(n,T))$ acting flags
and further we show the kernels of the corresponding homomorphisms are all
trivial. Thus, we give a positive answer to the Question 1.1.

##  2 Preliminaries

Let $\mathcal{F}^{T}_{n}$ denote the set of all flags of type $T$ of $[n]$. In
other words, $\mathcal{F}^{T}_{n}$ is the vertex set of $\Gamma(n,T)$.
Moreover, we set $\mathcal{F}^{(T|A)}_{n}=\\{f\in\mathcal{F}^{T}_{n}:~{}A\in
f\\}$, where $A\subseteq[n]$ and $|A|\in T$. Let $f$ and $g$ be two flags in
$\mathcal{F}^{T}_{n}$. If there exists an edge between $f$ and $g$ in
$\Gamma(n,T)$, then $f$ and $g$ are called _neighbor_ (see [10]). In addition,
$N(f)$ stands for the collection of all neighbours of vertex $f$. Here, we
state a well-known fact that is the key idea of solving Question 1.1.

######  Fact 2.1.

Let $f,g\in\mathcal{F}^{T}_{n}$. Then $|N(f)^{\alpha}|=|N(f^{\alpha})|$ and
$|N(f)\cap N(g)|=|N(f^{\alpha})\cap N(g^{\alpha})|$ for every $\alpha\in
Aut(\Gamma(n,T))$.

For convenience, we set $N(f,g)=N(f)\cap N(g)$ and
$N_{m}(n|T)={\rm{max}}\\{|N(f,g)|:f,g\in\mathcal{F}^{T}_{n}\\}$. Next, we
characterize $N_{m}(n|T)$ in some situations.

######  Proposition 2.2.

Let $f=\\{A,B\\}$ and $g=\\{C,D\\}$ be two flags in $\mathcal{F}^{T}_{n}$ with
$|A|=|C|=a$ and $|B|=|D|=b$, where $T=\\{a,b\\}$ with $a+b+1\leq n$ and
$a<\frac{n}{2}<b$. Then the followings hold.

(i) If $b>\frac{2n}{3}$, then $|N(f,g)|=N_{m}(n|T)$ if and only if $A=C$ and
$|B\cap D|=b-1$. In this case, $N_{m}(n|T)=C^{a}_{n-b-1}C^{2b-n-1}_{b-a-1}$.

(ii) If $b<\frac{2n}{3}$, then $|N(f,g)|=N_{m}(n|T)$ if and only if $B=D$ and
$|A\cap C|=a-1$. In this case, $N_{m}(n|T)=C^{a}_{n-b}C^{2b-n}_{b-a-1}$.

(iii) If $b=\frac{2n}{3}$, then $|N(f,g)|=N_{m}(n|T)$ if and only if either
$A=C$, $|B\cap D|=b-1$ or $B=D$, $|A\cap C|=a-1$. In this case,
$N_{m}(n|T)=C^{a}_{n-b-1}C^{2b-n-1}_{b-a-1}=C^{a}_{n-b}C^{2b-n}_{b-a-1}$.

Proof Let $h=\\{G,H\\}\in\mathcal{F}^{T}_{n}$ with $|G|=a$ and $|H|=b$. Since
$a+b+1\leq n$ and $a<\frac{n}{2}<b$, it follows that $h\in N(f,g)$ if and only
if $H\cap A=\emptyset$, $H\cap C=\emptyset$, $H\cup B=H\cup D=[n]$, $G\cap
B=\emptyset$ and $G\cap D=\emptyset$. In other words, $h\in N(f,g)$ if and
only if $G\subseteq[n]\setminus(B\cup D)$ and $\overline{B\cap D}\subseteq H$
and $H\cap(A\cup C)=\emptyset$. Clearly, the number of the choices of $G$ is
$C^{a}_{n-|B\cup D|}$. Consider $H$. Due to $\overline{B\cap D}\subseteq H$
and $H\cap(A\cup C)=\emptyset$, it follows that $A\cup C\subseteq B\cap D$ and
otherwise $N(f,g)=\emptyset$. Note that the number of the choices of $H$ is
equal to the number of the choices of $(b-|\overline{B\cap D}|)$-subsets from
$(B\cap D)\setminus(A\cup C)$. Therefore, the number of the choices of $H$ is
$C^{b-|\overline{B\cap D}|}_{|B\cap D|-|A\cup C|}$. Additionally, it is clear
that $|B\cup D|=2b-|B\cap D|$ and $|\overline{B\cap D}|=n-|B\cap D|$. Thus, we
have deduced that

$|N(f,g)|=C^{a}_{n-2b+|B\cap D|}C^{b+|B\cap D|-n}_{|B\cap D|-|A\cup
C|}=C^{a}_{n-2b+|B\cap D|}C^{n-b-|A\cup C|}_{|B\cap D|-|A\cup C|}.$ (2.1)

Here, we make a statement that every abnormal situation occurs imply
$N(f,g)=\emptyset$, where the abnormal situation is $|B\cap D|-|A\cup C|<0$ or
$n-2b+|B\cap D|<a$ and so on. Our goal is to find $N_{m}(n|T)$, and so we do
not discuss when $N(f,g)=\emptyset$ holds.

Fixing $|A\cup C|\geq a$. The equality 2.1 shows that $|N(f,g)|$ increases
with the increase of $|B\cap D|$. Likewise, fixing $|B\cap D|\leq b$,
$|N(f,g)|$ decreases with the increase of $|A\cup C|$. Therefore, $N_{m}(n|T)$
occurs in two possible situations. One is that $|A\cup C|=a$ and $|B\cap
D|=b-1$, and the other is that $|A\cup C|=a+1$ and $|B\cap D|=b$. If $|A\cup
C|=a$ and $|B\cap D|=b-1$, then by equality 2.1 we deduce that

$|N(f,g)|=C^{a}_{n-b-1}C^{2b-n-1}_{b-a-1}.$ (2.2)

If $|A\cup C|=a+1$ and $|B\cap D|=b$, then by equality 2.1 we infer that

$|N(f,g)|=C^{a}_{n-b}C^{2b-n}_{b-a-1}.$ (2.3)

Compare the equalities ( 2.2 ) and ( 2.3 ). We see that $\frac{(\
\ref{eq11}~{})}{(\
\ref{eq2}~{})}=\frac{C^{a}_{n-b-1}C^{2b-n-1}_{b-a-1}}{C^{a}_{n-b}C^{2b-n}_{b-a-1}}=\frac{2b-n}{n-b}$.
Obviously, if $b>\frac{2n}{3}$, then $\frac{\ \ref{eq11}}{\ \ref{eq2}}>1$ and
so $N_{m}(n|T)=C^{a}_{n-b-1}C^{2b-n-1}_{b-a-1}$. In this case,
$|N(f,g)|=N_{m}(n|T)$ if and only if $A=C$ and $|B\cap D|=b-1$. If
$b<\frac{2n}{3}$, then $\frac{\ \ref{eq11}}{\ \ref{eq2}}<1$ and thus
$N_{m}(n|T)=C^{a}_{n-b}C^{2b-n}_{b-a-1}$. In this case, $|N(f,g)|=N_{m}(n|T)$
if and only if $B=D$ and $|A\cap C|=a-1$. Additionally, if $b=\frac{2n}{3}$
then $\frac{\ \ref{eq11}}{\ \ref{eq2}}=1$ and thus $|N(f,g)|=N_{m}(n|T)$ if
and only if either $A=C$ and $|B\cap D|=b-1$ or $B=D$ and $|A\cap C|=a-1$. In
this case,
$N_{m}(n|T)=C^{a}_{n-b-1}C^{2b-n-1}_{b-a-1}=C^{a}_{n-b}C^{2b-n}_{b-a-1}$.
$\Box$

######  Remark 2.3.

Let $f,g\in\mathcal{F}^{T}_{n}$ such that $|N(f,g)|=N_{m}(n|T)$. Then
$|N(f^{\alpha},g^{\alpha})|=N_{m}(n|T)$ for any $\alpha\in Aut(\Gamma(n,T))$.

Proof This remark follows from Fact 2.1. $\Box$

By Proposition 2.2 (iii), we see that if $b=\frac{2n}{3}$ then $N_{m}(n|T)$
occurs in two cases. This urges us to further study this situation. Now we
investigate the second maximum of $|N(f,g)|$ and so we define
$N_{sm}(n|T)={\rm{max}}\\{|N(f,g)|:f,g\in\mathcal{F}^{T}_{n},|N(f,g)|<N_{m}(n|T)\\}$.

######  Proposition 2.4.

Let $f=\\{A,B\\}$ and $g=\\{C,D\\}$ be two flags in $\mathcal{F}^{T}_{n}$ with
$|A|=|C|=a$ and $|B|=|D|=b$, where $T=\\{a,b\\}$ with $a\leq\frac{n}{3}-1$ and
$b=\frac{2n}{3}$. Then, $|N(f,g)|=N_{sm}(n|T)$ if and only if $|A\cup C|=a+1$
and $|B\cup D|=b+1$.

Proof According to the equality 2.1, we see that $|N(f,g)|=N_{sm}(n|T)$ occurs
in three possible cases, those are, $|A\cup C|=a+1$, $|B\cup D|=b+1$ or
$|A\cup C|=a+2$, $|B\cup D|=b$ or $|A\cup C|=a$, $|B\cup D|=b+2$. Next, we
start to compute $|N(f,g)|$ in three cases respectively.

If $|A\cup C|=a+1$ and $|B\cup D|=b+1$, then $|B\cap D|=b-1$. In this case, by
equality 2.1, we infer that

$|N(f,g)|=C^{a}_{n-b-1}C^{2b-n-1}_{b-a-2}.$ (2.4)

Similarly, if $|A\cup C|=a$ and $|B\cup D|=b+2$, then

$|N(f,g)|=C^{a}_{n-b-2}C^{2b-n-2}_{b-a-2};$ (2.5)

and if $|A\cup C|=a+2$ and $|B\cup D|=b$ then

$|N(f,g)|=C^{a}_{n-b}C^{2b-n}_{b-a-2}.$ (2.6)

Compare three equalities ( 2.4 ) and ( 2.5 ) and ( 2.6 ). We deduce that

$\frac{(\ \ref{eq4}~{})}{(\
\ref{eq3}~{})}=\frac{C^{a}_{n-b-2}C^{2b-n-2}_{b-a-2}}{C^{a}_{n-b-1}C^{2b-n-1}_{b-a-2}}=\frac{(n-b-a-1)(2b-n-1)}{(n-b-1)(n-b-a)}=\frac{n-b-a-1}{n-b-a}<1~{}\rm{and}$
$\frac{(\ \ref{eq5}~{})}{(\
\ref{eq3}~{})}=\frac{C^{a}_{n-b}C^{2b-n}_{b-a-2}}{C^{a}_{n-b-1}C^{2b-n-1}_{b-a-2}}=\frac{(n-b)(n-b-a-1)}{(n-b-a)(2b-n)}=\frac{n-b-a-1}{n-b-a}<1.$

Therefore, $|N(f,g)|=N_{sm}(n|T)$ if and only if $|A\cup C|=a+1$ and $|B\cup
D|=b+1$. In particular, $N_{sm}(n|T)=C^{a}_{n-b-1}C^{2b-n-1}_{b-a-2}$. $\Box$

So far, we have seen that $N_{sm}(n|T)$ occurs in unique form when
$b=\frac{2n}{3}$. Let $f,g\in\mathcal{F}^{T}_{n}$ where $T=\\{a,b\\}$ with
$a\leq\frac{n}{3}-1$ and $b=\frac{2n}{3}$. Define
$SM(f)=\\{h\in\mathcal{F}^{T}_{n}:|N(f,h)|=N_{sm}(n|T)\\}$ and
$SM(f,g)=SM(f)\cap SM(g)$. Next we state two results which are useful in
dealing with the case that $b=\frac{2n}{3}$.

######  Proposition 2.5.

Suppose that $f,g\in\mathcal{F}^{T}_{n}$, where $T=\\{a,b\\}$ with
$a\leq\frac{n}{3}-1$ and $b=\frac{2n}{3}$. Then
$|SM(f,g)|=|SM(f^{\alpha},g^{\alpha})|$ for any $\alpha\in Aut(\Gamma(n,T))$.

Proof Assume that $h\in\mathcal{F}^{T}_{n}$ such that $h\in SM(f,g)$. Thus,
$|N(f,h)|=|N(g,h)|=N_{sm}(n|T)$. Then by Fact 2.1 we deduce that
$|N(f,h)|=|N(f^{\alpha},h^{\alpha})|$ and
$|N(g,h)|=|N(g^{\alpha},h^{\alpha})|$ for any $\alpha\in Aut(\Gamma(n,T))$.
Conversely, for any $\alpha\in Aut(\Gamma(n,T))$, if $v\in
SM(f^{\alpha},g^{\alpha})$ then $v^{\alpha^{-1}}\in SM(f,g)$. This completes
the proof of this proposition. $\Box$

######  Proposition 2.6.

Let $f=\\{A,B\\}$, $g=\\{A,C\\}$, $x=\\{D,E\\}$ and $y=\\{F,E\\}$ be four
flags in $\mathcal{F}^{T}_{n}$ such that $|B\cap C|=b-1$ and $|D\cap F|=a-1$,
where $T=\\{a,b\\}$ with $a\leq\frac{n}{3}-1$ and $b=\frac{2n}{3}$. Then
$|SM(f,g)|\neq|SM(x,y)|$.

Proof Let $h=\\{G,H\\}\in\mathcal{F}^{T}_{n}$. By applying Proposition 2.4, we
deduce that $h\in SM(f,g)$ if and only if $|G\cap A|=a-1$ and $|H\cap B|=b-1$
and $|H\cap C|=b-1$. Count $|SM(f,g)|$. Note that there exist two possible
shapes for $H$, those are, $B\cap C\subseteq H$ and $B\cap C\not\subseteq H$
respectively. In the case of $B\cap C\subseteq H$, we see that $H=(B\cap
C)\cup\\{i\\}$ where $i\in[n]\setminus(B\cup C)$ and so the number of the
choices of $H$ is $C^{1}_{n-b-1}$. Fix an $H$, if $G\subseteq B\cap C$ then
$G$ is the union of a $(a-1)$-subset in $A$ and a $1$-subset in $(B\cap
C)\setminus A$, and thus the number of the choices of $G$ is
$C^{a-1}_{a}C^{1}_{b-1-a}$; and if $G\not\subseteq B\cap C$ then $G$ is the
union of an $(a-1)$-subset in $A$ and $H\setminus(B\cap C)$ and so the number
of the choices of $G$ is $C^{a-1}_{a}$. Hence, in the case of $B\cap
C\subseteq H$, the number of the choices of $h$ in $SM(f,g)$ is

$C^{1}_{n-b-1}(C^{a-1}_{a}C^{1}_{b-1-a}+C^{a-1}_{a})=(\frac{n}{3}-1)a(\frac{2n}{3}-a-1)+(\frac{n}{3}-1)a.$

Consider $B\cap C\not\subseteq H$. In this case, $H=(B\setminus
C)\cup(C\setminus B)\cup K$ where $K$ is a $(b-2)$-subset of $B\cap C$. In
addition, there exist two possible subcases for $H$, those are, $A\subseteq H$
and $A\not\subseteq H$. In the subcase of $A\subseteq H$, the number of the
choices of $H$ is $C^{b-2-a}_{b-1-a}$. Fix an $H$, $G$ is the union of an
$(a-1)$-subset of $A$ and a $1$-subset of $H\setminus A$, and thus the number
of the choices of $G$ is $C^{a-1}_{a}C^{1}_{b-a}$. If $A\not\subseteq H$, then
$H=(B\cup C)\setminus\\{j\\}$ where $j\in A$, and so the number of the choices
of $H$ is $C^{1}_{a}$. Fix an $H$, $G=(A\setminus\\{j\\})\cup\\{k\\}$ where
$k\in H\setminus A$ and so the number of the choices of $G$ is
$C^{1}_{b-a+1}$. Therefore, in the case of $B\cap C\not\subseteq H$, the
number of the choices of $h$ in $SM(f,g)$ is

$C^{b-2-a}_{b-1-a}C^{a-1}_{a}C^{1}_{b-a}+C^{1}_{a}C^{1}_{b-a+1}=(\frac{2n}{3}-a-1)a(\frac{2n}{3}-a)+a(\frac{2n}{3}-a+1).$

So we deduce that

$|SM(f,g)|=a(\frac{2n}{3}-a)(n-a-1)+a.$

Likewise, we can count $|SM(x,y)|$. Let $z=\\{U,V\\}\in\mathcal{F}^{T}_{n}$.
By applying Proposition 2.4, we deduce that $z\in SM(x,y)$ if and only if
$|U\cap D|=a-1$ and $|U\cap F|=a-1$ and $|V\cap E|=b-1$. Note that there exist
two possible shapes for $U$, those are, $U\subseteq E$ and $U\not\subseteq E$
respectively. Consider $U\subseteq E$. If $D\cap F\subseteq U$, then $U=(D\cap
F)\cup\\{p\\}$ where $p\in E\setminus(D\cup F)$, and so the number of the
choices of $U$ is $C^{1}_{b-1-a}$. Fix an $U$, $V$ is the union of a
$(b-1)$-subset containing $U$ of $E$ and a $1$-subset of $[n]\setminus E$, and
thus the number of the choices of $V$ is $C^{b-a-1}_{b-a}C^{1}_{n-b}$. If
$D\cap F\not\subseteq U$, then $U=(D\cup F)\setminus\\{q\\}$ where $q\in D\cap
F$ and so the number of the choices of $U$ is $C^{1}_{a-1}$. Fix an $U$, the
number of the choices of $V$ is $C^{b-a-1}_{b-a}C^{1}_{n-b}$ too. Therefore,
in the case of $U\subseteq E$, the number of the choices of $z$ in $SM(x,y)$
is

$C^{1}_{b-1-a}C^{b-a-1}_{b-a}C^{1}_{n-b}+C^{1}_{a-1}C^{b-a-1}_{b-a}C^{1}_{n-b}=(\frac{2n}{3}-a-1)(\frac{2n}{3}-a)\frac{n}{3}+(a-1)(\frac{2n}{3}-a)\frac{n}{3}.$

Consider $U\not\subseteq E$. In this case, $U=(D\cap F)\cup\\{r\\}$ where
$r\in[n]\setminus E$ and so the number of the choices of $U$ is $C^{1}_{n-b}$.
Fix an $U$, $V$ is the union of $U$ and a $(b-a)$-subset of $E\setminus(D\cap
F)$ and thus the number of the choices of $V$ is $C^{b-a}_{b-a+1}$. Hence, in
the case of $U\not\subseteq E$, the number of the choices of $z$ in $SM(x,y)$
is

$C^{1}_{n-b}C^{b-a}_{b-a+1}=\frac{n}{3}(\frac{2n}{3}-a+1).$

Therefore, we deduce that

$|SM(x,y)|=\frac{n}{3}(\frac{2n}{3}-a)(\frac{2n}{3}-1)+\frac{n}{3}.$

By computing, we derive
$|SM(f,g)|-|SM(x,y)|=(a-\frac{n}{3})\\{1-(\frac{2n}{3}-a)(a-\frac{2n}{3}+1)\\}\neq
0$ for all $a\leq\frac{n}{3}-1$. The proof of this proposition is complete.
$\Box$

Now we end this section by introducing a graph that will be used. Let
$\dbinom{[n]}{k}$ denote the set of all $k$-subsets of $[n]$, where $1\leq
k\leq n-1$. If $A,B\in\dbinom{[n]}{k}$ such that $|A\cap B|=k-1$, then we say
that $A$ and $B$ are _almost identical_. Define _almost identical graph_
$AIG(n,k)$ has as vertex set $\dbinom{[n]}{k}$ with two vertices being
adjacent when the corresponding subsets are almost identical.

######  Proposition 2.7.

Let $n,k$ be positive integers with $1\leq k\leq n-1$. Then $AIG(n,k)$ is a
connected graph.

Proof Let $A$ and $B$ be two vertices of $AIG(n,k)$. If $|A\cap B|=k-1$ then
$A$ and $B$ are adjacent. So we assume that
$A=\\{a_{1},a_{2},...,a_{l},b_{1},b_{2},...,b_{k-l}\\}$ and
$B=\\{a_{1},a_{2},...,a_{l},c_{1},c_{2},...,c_{k-l}\\}$ with $|A\cap
B|=|\\{a_{1},a_{2},...,a_{l}\\}|=l<k-1$. We note the characteristic that

$\\{b_{1},b_{2},...,b_{k-l}\\}\leftrightarrow\\{b_{1},b_{2},...,b_{k-l-1},c_{1}\\}\leftrightarrow\\{b_{1},b_{2},...,b_{k-l-2},c_{1},c_{2}\\}\leftrightarrow\cdot\cdot\cdot\leftrightarrow\\{b_{1},c_{1},c_{2},...,c_{k-l-1}\\},$

which implies that there exists a path between $A$ and $B$. The proof of this
proposition is complete. $\Box$

##  3 Main Result

It is well-known that if $|T|=1$ then $\Gamma(n,T)$ is isomorphic to a Kneser
graph. However, we note that if $T=\\{t\\}$ with $t>\frac{n}{2}$ then
$\Gamma(n,T)$ does not fit the traditional definition of Kneser graph. In
order to facilitate the readers, we first give a proof of this well-known
result.

######  Lemma 3.1.

Let $T=\\{t\\}\subseteq[n-1]$. Then $Aut(\Gamma(n,T))\cong S_{n}$.

Proof Let $T=\\{t\\}\subseteq[n-1]$. If $t\leq\frac{n}{2}$, then by [5,
Corollary 7.8.2] we see that $Aut(\Gamma(n,T))\cong S_{n}$. Suppose
$t>\frac{n}{2}$. Define a map $\sigma:\Gamma(n,T)\rightarrow
KG(n,n-t),A\mapsto\overline{A}$, where $\overline{A}$ denotes the complement
of $A$ in $[n]$. It is easy to verify that $\sigma$ is an isomorphic map from
$\Gamma(n,T)$ to $KG(n,n-t)$. The proof of this lemma is complete. $\Box$

######  Proposition 3.2.

Let $T=\\{a,b\\}\subseteq[n-1]$ such that $a<\frac{n}{2}$ and $\frac{2n}{3}<b$
and $a+b+1\leq n$. Then
$\Sigma=\\{\mathcal{F}^{(T|A)}_{n}:A\subseteq[n],|A|=a\\}$ is a system of
nontrivial blocks of $Aut(\Gamma(n,T))$ acting on $\mathcal{F}^{T}_{n}$.

Proof Let $\rho$ be an automorphism in $Aut(\Gamma(n,T))$. Pick $f=\\{A,B\\}$
and $g=\\{A,C\\}$ in $\mathcal{F}^{(T|A)}$ such that $|B\cap C|=b-1$ and
$|A|=a$. It follows from Proposition 2.2 (i) and Remark 2.3 that there exists
an $a$-subset $D\subseteq[n]$ such that
$f^{\rho},g^{\rho}\in\mathcal{F}^{(T|D)}$. In particular, if
$f^{\rho}=\\{D,G\\}$ and $g^{\rho}=\\{D,H\\}$ then $|G\cap H|=b-1$. By the
same token, we deduce that $h^{\rho}\in\mathcal{F}^{(T|D)}$ in case when
$h=\\{A,X\\}\in\mathcal{F}^{(T|A)}$ with $|X\cap B|=b-1$ or $|X\cap C|=b-1$.
Continue moving forward along this line of thought, Proposition 2.7 indicates
that $h^{\rho}\in\mathcal{F}^{(T|D)}$ for every $h\in\mathcal{F}^{(T|A)}$.
Therefore, $\sum$ is a system of nontrivial blocks of $Aut(\Gamma(n,T))$
acting on $\mathcal{F}^{T}_{n}$. $\Box$

######  Lemma 3.3.

Let $T=\\{a,b\\}\subseteq[n-1]$ such that $a<\frac{n}{2}$ and $\frac{2n}{3}<b$
and $a+b+1\leq n$. Then $Aut(\Gamma(n,T))\cong S_{n}$.

Proof It follows from Proposition 3.2 that
$\sum=\\{\mathcal{F}^{(T|A)}_{n}:A\subseteq[n],|A|=a\\}$ is a system of blocks
of $Aut(\Gamma(n,T))$ acting on $\mathcal{F}^{T}_{n}$. Consider the induced
action of $Aut(\Gamma(n,T))$ acting on $\sum$, that is,
$Aut(\Gamma(n,T))|_{\sum}$. Let $\rho\in Aut(\Gamma(n,T))$ and
${\mathcal{F}}^{(T|A)}_{n},{\mathcal{F}}^{(T|B)}_{n},{\mathcal{F}}^{(T|C)}_{n},{\mathcal{F}}^{(T|D)}_{n}\in\Sigma$
such that ${{\mathcal{F}}^{(T|A)}_{n}}^{\rho}={{\mathcal{F}}^{(T|B)}_{n}}$ and
${{\mathcal{F}}^{(T|C)}_{n}}^{\rho}={{\mathcal{F}}^{(T|D)}_{n}}$. Note that
$B\cap D=\emptyset$ if and only if $A\cap C=\emptyset$. Otherwise one of
$\\{(f,g),(f^{\rho},g^{\rho})\\}$ is an edge and the other is not an edge for
some $f\in\mathcal{F}^{(T|A)}_{n}$ and $g\in\mathcal{F}^{(T|C)}_{n}$.
Therefore, $Aut(\Gamma(n,T))|_{\Sigma}\leq Aut(KG(n,a))$. On the other hand,
the symmetric group on the set $[n]$ induces an automorphism group of
$\Gamma(n,T)$, and thus $Aut(\Gamma(n,T))|_{\Sigma}\cong Aut(KG(n,a))\cong
S_{n}$. Thus, it suffices to check that no nonidentity automorphism can fix
all the blocks of $\sum$. Let $\rho$ be an automorphism in $Aut(\Gamma(n,T))$
such that ${{\mathcal{F}}^{(T|A)}_{n}}^{\rho}={{\mathcal{F}}^{(T|A)}_{n}}$ for
all $A\subseteq[n]$ with $|A|=a$. Assume that there exist two distinct flags
$f=\\{A,B\\}$ and $g=\\{A,C\\}$ in $\mathcal{F}^{(T|A)}_{n}$ such that
$f^{\rho}=g$ for some $A\subseteq[n]$ with $|A|=a$. Since $f\neq g$, it
follows that $B\neq C$, in other words, there exists an $i\in C\setminus B$.
Hence, there exists a flag $h=\\{D,E\\}\in N(f)$ such that $i\in D$ and
$|D|=a$. By Fact 2.1, we see that $N(f)^{\rho}=N(g)$. However, it is clear
that $N(g)\cap\mathcal{F}^{(T|D)}_{n}=\emptyset$, and which indicates that
${{\mathcal{F}}^{(T|D)}_{n}}^{\rho}\neq{{\mathcal{F}}^{(T|D)}_{n}}$, a
contradiction. $\Box$

######  Proposition 3.4.

Let $T=\\{a,b\\}\subseteq[n-1]$ such that $a<\frac{n}{2}<b<\frac{2n}{3}$ and
$a+b+1\leq n$. Then $\Omega=\\{\mathcal{F}^{(T|B)}_{n}:B\subseteq[n],|B|=b\\}$
is a system of nontrivial blocks of $Aut(\Gamma(n,T))$ acting on
$\mathcal{F}^{T}_{n}$.

Proof Let $\rho$ be an automorphism in $Aut(\Gamma(n,T))$. Take two flags
$f=\\{A,B\\}$ and $g=\\{C,B\\}$ in $\mathcal{F}^{(T|B)}$ with $|A\cap C|=a-1$
and $|B|=b$. It follows from Proposition 2.2 (ii) and Remark 2.3 that there
exists a $b$-subset $D\subseteq[n]$ such that
$f^{\rho},g^{\rho}\in\mathcal{F}^{(T|D)}$. Additionally, if
$f^{\rho}=\\{G,D\\}$ and $g^{\rho}=\\{H,D\\}$ then $|G\cap H|=a-1$. In a
similar manner, $h^{\rho}\in\mathcal{F}^{(T|D)}$ in case when
$h=\\{X,B\\}\in\mathcal{F}^{(T|B)}$ with $|X\cap A|=a-1$ or $|X\cap C|=a-1$.
Along this idea of thought, Proposition 2.7 implies that
$h^{\rho}\in\mathcal{F}^{(T|D)}$ for every $h\in\mathcal{F}^{(T|B)}$.
Therefore, $\Omega$ is a system of nontrivial blocks of $Aut(\Gamma(n,T))$
acting on $\mathcal{F}^{T}_{n}$. $\Box$

######  Lemma 3.5.

Let $T=\\{a,b\\}\subseteq[n-1]$ such that $a<\frac{n}{2}<b<\frac{2n}{3}$ and
$a+b+1\leq n$. Then $Aut(\Gamma(n,T))\cong S_{n}$.

Proof By Proposition 3.4, it follows that
$\Omega=\\{\mathcal{F}^{(T|B)}_{n}:B\subseteq[n],|B|=b\\}$ is a system of
nontrivial blocks of $Aut(\Gamma(n,T))$ acting on $\mathcal{F}^{T}_{n}$.
Consider the induced action $Aut(\Gamma(n,T))|_{\Omega}$ of $Aut(\Gamma(n,T))$
acting on $\Omega$. Let $\rho\in Aut(\Gamma(n,T))$ and
${\mathcal{F}}^{(T|A)}_{n},{\mathcal{F}}^{(T|B)}_{n},{\mathcal{F}}^{(T|C)}_{n},{\mathcal{F}}^{(T|D)}_{n}\in\Omega$
such that ${{\mathcal{F}}^{(T|A)}_{n}}^{\rho}={{\mathcal{F}}^{(T|B)}_{n}}$ and
${{\mathcal{F}}^{(T|C)}_{n}}^{\rho}={{\mathcal{F}}^{(T|D)}_{n}}$. Note that
$B\cup D=[n]$ if and only if $A\cup C=[n]$. Otherwise one of
$(f,g),(f^{\rho},g^{\rho})$ is an edge and the other is not an edge will occur
for some $f\in\mathcal{F}^{(T|A)}_{n}$ and $g\in\mathcal{F}^{(T|C)}_{n}$.
Therefore, $Aut(\Gamma(n,T))|_{\Omega}\leq Aut(KG(n,n-b))$. On the other hand,
the symmetric group on the set $[n]$ induces an automorphism group of
$\Gamma(n,T)$, and thus $Aut(\Gamma(n,T))|_{\Omega}\cong Aut(KG(n,n-b))\cong
S_{n}$. Thus, it suffices to check that no nonidentity automorphism can fix
all the blocks of $\Omega$. Let $\rho$ be an automorphism in
$Aut(\Gamma(n,T))$ such that
${{\mathcal{F}}^{(T|B)}_{n}}^{\rho}={{\mathcal{F}}^{(T|B)}_{n}}$ for all
$B\subseteq[n]$ with $|B|=b$. Assume that there exist two distinct flags
$f=\\{A,B\\}$ and $g=\\{C,B\\}$ in $\mathcal{F}^{(T|B)}_{n}$ such that
$f^{\rho}=g$ for some $B\subseteq[n]$ with $|B|=b$. Since $f\neq g$, it
follows that $A\neq C$, in other words, there exists an $i\in C\setminus A$.
Hence, there exists a flag $h=\\{D,E\\}\in N(f)$ such that $i\in D$ and
$|D|=a$ and $|E|=b$. By Fact 2.1, we see that $N(f)^{\rho}=N(g)$. However, it
is clear that $N(g)\cap\mathcal{F}^{(T|E)}_{n}=\emptyset$, and which indicates
that ${{\mathcal{F}}^{(T|E)}_{n}}^{\rho}\neq{{\mathcal{F}}^{(T|E)}_{n}}$, a
contradiction. $\Box$

######  Lemma 3.6.

Let $T=\\{a,b\\}\subseteq[n-1]$ such that $a\leq\frac{n}{3}-1$ and
$b=\frac{2n}{3}$. Then the followings hold.

(i) $\Sigma=\\{\mathcal{F}^{(T|A)}_{n}:A\subseteq[n],|A|=a\\}$ and
$\Omega=\\{\mathcal{F}^{(T|B)}_{n}:B\subseteq[n],|B|=b\\}$ are two systems of
nontrivial blocks of $Aut(\Gamma(n,T))$ acting on $\mathcal{F}^{T}_{n}$.

(ii) $Aut(\Gamma(n,T))\cong S_{n}$.

Proof Let $\rho$ be an automorphism in $Aut(\Gamma(n,T))$. Pick $f=\\{A,B\\}$
and $g=\\{A,C\\}$ in $\mathcal{F}^{(T|A)}_{n}$ such that $|B\cap C|=b-1$ and
$|A|=a$. It follows from Proposition 2.2 (iii) and Remark 2.3 and Proposition
2.5 and Proposition 2.6 that there exists a $a$-subset $D\subseteq[n]$ such
that $f^{\rho},g^{\rho}\in\mathcal{F}^{(T|D)}$. An argument similar to the one
used in the proof of Proposition 3.2 shows that $\Sigma$ is a systems of
nontrivial blocks of $Aut(\Gamma(n,T))$ acting on $\mathcal{F}^{T}_{n}$. In a
similar manner, we can show that $\Omega$ is also a systems of nontrivial
blocks of $Aut(\Gamma(n,T))$ acting on $\mathcal{F}^{T}_{n}$. Proceeding as in
the proof of Lemma 3.3 or Lemma 3.5, we have $Aut(\Gamma(n,T))\cong S_{n}$.
$\Box$

So far, we have not used [8, Remark 5.13] to give a positive answer to
Question 1.1. On the other hand, by applying [8, Lemma 2.1(b)], we obtain the
following theorem.

######  Theorem 3.7.

Let $T=\\{a,b\\}\subseteq[n-1]$ with $a<\frac{n}{2}<b$ and $a+b\neq n$. Then
$Aut(\Gamma(n,T))\cong S_{n}$.

##  4 Concluding Remarks

Let $T=\\{a,b\\}\subseteq[n-1]$ with $a<b$. Note that three situations are
left to consider, those are, $a+b=n$ and $b\leq\frac{n}{2}$ and
$\frac{n}{2}\leq a$.

Let $T=\\{a,b\\}\subseteq[n-1]$ with $a<b$ and $a+b=n$. Suppose that
$f=\\{A,B\\}$ and $g=\\{C,D\\}$ are two flags in $\mathcal{F}^{T}_{n}$ such
that $|A|=|C|=a$ and $|B|=|D|=b$. Clearly, $(f,g)$ is an edge if and only if
$C=\overline{B}$ and $D=\overline{A}$. Define $\Delta^{T(A|B)}_{n}=\\{f,g\\}$
where $f=\\{A,B\\}$ and $g=\\{\overline{A},\overline{B}\\}$ are in
$\mathcal{F}^{T}_{n}$, and $\Omega=\\{\Delta^{T(A|B)}_{n}:A\subset
B\subset[n],|A|=a,|B|=b\\}$. It is straightforward to see that $\Omega$ is a
system of nontrivial blocks of $Aut(\Gamma(n,T))$ acting on
$\mathcal{F}^{T}_{n}$, and further the following lemma holds.

######  Lemma 4.1.

Let $\Omega=\\{\Delta^{T(A|B)}_{n}:A\subset B\subset[n],|A|=a,|B|=b\\}$, where
$T=\\{a,b\\}\subseteq[n-1]$ with $a<b$ and $a+b=n$. Then
$Aut(\Gamma(n,T))\cong N\wr S_{m}$, where
$N=\overbrace{S_{2}\times\cdot\cdot\cdot\times S_{2}}^{m}$ and
$m=|\Omega|=\frac{1}{2}C^{b}_{n}C^{a}_{b}$.

For the case that $b\leq\frac{n}{2}$, we consider a more general cases, as
follows.

######  Lemma 4.2.

Let $\Sigma=\\{\mathcal{F}^{(T|A)}_{n}:A\subseteq[n],|A|=t_{r}\\}$, where
$T=\\{t_{1},t_{2},...,t_{r}\\}\subseteq[n-1]$ such that
$t_{1}<t_{2}<\cdot\cdot\cdot<t_{r}\leq\frac{n}{2}$. Then $\Sigma$ is a system
of nontrivial blocks of $Aut(\Gamma(n,T))$ acting on $\mathcal{F}^{T}_{n}$. In
addition, $Aut(\Gamma(n,T))\cong N\wr S_{n}$ where
$N=\overbrace{S_{m}\times\cdot\cdot\cdot\times S_{m}}^{C^{t_{r}}_{n}}$ and
$m=|\mathcal{F}^{(T|A)}_{n}|$ for a $\mathcal{F}^{(T|A)}_{n}\in\Sigma$.

Proof Let $\rho\in Aut(\Gamma(n,T))$ and $\mathcal{F}^{(T|A)}_{n}\in\Sigma$.
Suppose that $f\in\mathcal{F}^{(T|A)}$ and $f^{\rho}\in\mathcal{F}^{(T|B)}$
for some $\mathcal{F}^{(T|B)}\in\Sigma$. It suffices to prove that
$g^{\rho}\in\mathcal{F}^{(T|B)}$ for any $g\in\mathcal{F}^{(T|A)}$. Proof by
contradiction. Assume that there exists a $h\in\mathcal{F}^{(T|A)}$ such that
$h^{\rho}\in\mathcal{F}^{(T|C)}\neq\mathcal{F}^{(T|B)}$. It is clear that

$N(f,h)=\bigcup\mathcal{F}^{(T|D)}_{n},~{}{\rm{where}}~{}D\subseteq[n]\setminus
A~{}{\rm{with}}~{}|D|=t_{r}.$

Likewise, it is simple to see that

$N(f^{\rho},h^{\rho})=\bigcup\mathcal{F}^{(T|D)},~{}{\rm{where}}~{}D\subseteq[n]\setminus(B\cup
C)~{}{\rm{with}}~{}|D|=t_{r}.$

Obviously, $|N(f^{\rho},h^{\rho})|<|N(f,h)|$, and which is contradict to Fact
2.1. Therefore, $\Sigma$ is a system of nontrivial blocks of
$Aut(\Gamma(n,T))$ acting on $\mathcal{F}^{T}_{n}$.

Let $\rho\in Aut(\Gamma(n,T))$ and
${\mathcal{F}}^{(T|A)}_{n},{\mathcal{F}}^{(T|B)}_{n},{\mathcal{F}}^{(T|C)}_{n},{\mathcal{F}}^{(T|D)}_{n}\in\Sigma$
such that ${{\mathcal{F}}^{(T|A)}_{n}}^{\rho}={{\mathcal{F}}^{(T|B)}_{n}}$ and
${{\mathcal{F}}^{(T|C)}_{n}}^{\rho}={{\mathcal{F}}^{(T|D)}_{n}}$. Clearly,
$B\cap D=\emptyset$ if and only if $A\cap C=\emptyset$. Hence, we deduce that
$Aut(\Gamma(n,T))|_{\Sigma}\leq Aut(KG(n,t_{r}))$. Additionally, the symmetric
group on the set $[n]$ induces an automorphism group of $\Gamma(n,T)$, and
therefore $Aut(\Gamma(n,T))|_{\Sigma}\cong Aut(KG(n,t_{r}))\cong S_{n}$. On
the other hand, it is obvious that the kernel of the natural homomorphism from
$Aut(\Gamma(n,T))$ to $Aut(\Gamma(n,T))|_{\Sigma}$ is the direct product of
the symmetric groups on all blocks of $\Sigma$, and therefore
$Aut(\Gamma(n,T))\cong N\wr S_{n}$ where
$N=\overbrace{S_{m}\times\cdot\cdot\cdot\times S_{m}}^{C^{t_{r}}_{n}}$ and
$m=|\mathcal{F}^{(T|A)}_{n}|$ for a $\mathcal{F}^{(T|A)}_{n}\in\Sigma$. $\Box$

By [8, Lemma 2.1(b)] and Lemma 4.2, we derive the following corollary.

######  Corollary 4.3.

Let $T=\\{t_{1},t_{2},...,t_{r}\\}\subseteq[n-1]$ with $\frac{n}{2}\leq
t_{1}<t_{2}<\cdot\cdot\cdot<t_{r}$. Then $Aut(\Gamma(n,T))\cong N\wr S_{n}$,
where $N=\overbrace{S_{m}\times\cdot\cdot\cdot\times S_{m}}^{C^{t_{1}}_{n}}$
and $m=|\mathcal{F}^{(T|A)}_{n}|$ with $|A|=n-t_{1}$.

##  5 Acknowledgement

We are very grateful to the anonymous referees for their useful suggestions
and comments.

##  References

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# A severe challenge to the MOND phenomenology in our Galaxy

Man Ho Chan & Ka Chung Law  Department of Science and Environmental Studies,
The Education University of Hong Kong, Hong Kong, China<EMAIL_ADDRESS>
###### Abstract

Modified Newtonian Dynamics (MOND) is one of the most popular alternative
theories of dark matter to explain the missing mass problem in galaxies.
Although it remains controversial regarding MOND as a fundamental theory, MOND
phenomenology has been shown to widely apply in different galaxies, which
gives challenges to the standard $\Lambda$ cold dark matter model. In this
article, we derive analytically the galactic rotation curve gradient in the
MOND framework and present a rigorous analysis to examine the MOND
phenomenology in our Galaxy. By assuming a benchmark baryonic disk density
profile and two popular families of MOND interpolating functions, we show for
the first time that the recent discovery of the declining Galactic rotation
curve in the outer region ($R\approx 17-23$ kpc) can almost rule out the MOND
phenomenology at more than $5\sigma$. This strongly supports some of the
previous studies claiming that MOND is neither a fundamental theory nor a
universal description of galactic properties.

Modified Gravity, Galaxy

##  1 Introduction

Dark matter theory and modified gravity theory are two kinds of competing
theories to explain the missing mass problem in galaxies and galaxy clusters
(Bertone & Tait, 2018). The controversy between these two theories is still
on-going (Martens et al., 2022). The successful predictions based on the
standard $\Lambda$ cold dark matter ($\Lambda$CDM) model in large-scale
structures have gained important credence on dark matter theory (Croft et al.,
1999; Spergel & Steinhardt, 2000; Peacock et al., 2001) while the galactic
relations (e.g. baryonic Tully-Fisher relation, radial acceleration relation)
show a certain support to modified gravity theory (McGaugh, 2020; Islam &
Dutta, 2020).

As one of the earliest versions of modified gravity theory, Modified Newtonian
Dynamics (MOND) has been examined for four decades. It is well-known that MOND
works very well in galaxies but it gives poor fits with the data of galaxy
clusters (Sanders & McGaugh, 2002; Chan & Del Popolo, 2020). Although it is
dubious for MOND to be a fundamental theory (Chan, 2013), the phenomenological
description using MOND agrees with the data in galaxies. For example, the
baryonic Tully-Fisher relation (Lelli et al., 2019) and the radial
acceleration relation (McGaugh, Lelli & Schombert, 2016) in galaxies give
excellent agreement with the MOND’s predictions (McGaugh, 2020; Islam & Dutta,
2020). Also, the apparent flat rotation curves (RCs) in most of the galaxies
match the MOND behavior at large distance from the galactic centres (Sanders &
McGaugh, 2002; Gentile, 2008). Therefore, these potentially suggest that MOND
can be regarded as a universal phenomenological description in galaxies (the
MOND phenomenology) (Gentile, 2008; Dutton et al., 2019). Many studies have
used the MOND phenomenology to study galactic dark matter and galactic scaling
relations (Gentile et al., 2009; Ho, Minic & Ng, 2010). However, most of the
galactic data have large uncertainties. For example, the high-quality rotation
curve data (SPARC) used in supporting MOND have an average of 7.1%
observational uncertainty (not including systematic uncertainties) (Lelli,
McGaugh & Schombert, 2016). Therefore, it is very difficult for us to
rigorously verify or falsify MOND as a universal phenomenological description
on galactic scale.

On the other hand, recent high-quality observations of the Milky Way rotation
curve (MWRC) data in the outer region ($R>17$ kpc) give very small marginal
errors (with an average of 2-3% observational uncertainty only) in the
rotation curve fittings (Eilers et al., 2019; Wang, Hammer & Yang, 2022;
Labini et al., 2023; Ou et al., 2023). The MWRC shows a very clear decreasing
trend from $V\approx 220$ km/s to $V\approx 170$ km/s in the range of
$R=12-27$ kpc (Labini et al., 2023; Ou et al., 2023). The decreasing RC may
even extend to 200 kpc based on the data of the Milky Way satellites, although
the decreasing rate is relatively smaller than that in the range of $R=12-27$
kpc (Vasiliev, Belokurov & Erkal, 2021; Wang, Hammer & Yang, 2022). Since MOND
generally suggests a flat rotation curve for large $R$, such a decline in MWRC
is not trivial for MOND. However, statistical margins might still allow an
almost flat rotation curve to fit the decline in MWRC due to the measurement
uncertainties.

In view of this problem, a more precise way is to focus particularly on the RC
gradient $dV/dR$ to investigate MOND. Earlier measurements give
$dV/dR=-(1.7\pm 0.1)$ km/s/kpc at large $R$ (Eilers et al., 2019), which can
still be explained by the MOND behavior (McGaugh, 2018, 2023). However, more
recent measurements and analyses give $dV/dR=-(2.3\pm 0.2)$ km/s/kpc (Wang et
al., 2023) and even $dV/dR<-3$ km/s/kpc (Labini et al., 2023; Ou et al., 2023)
at large $R$, which might give challenges to the MOND phenomenological
description.

In this article, we present a rigorous analysis to understand how large of
$dV/dR$ can MOND tolerate in galaxies. We specifically use the MWRC data to
examine the MOND phenomenology using two major families of MOND interpolating
functions. By assuming a benchmark baryonic disk density profile, we show that
the MWRC data can rule out MOND phenomenology for the two major families of
interpolating functions at more than $5\sigma$, which can almost exhaustively
represent all possible variations of MOND. This suggests that there is likely
no room for MOND phenomenology in our Galaxy, which provides the first severe
challenge to the alleged universal nature of MOND phenomenology on galactic
scale.

##  2 The rotation curve behavior in the MOND framework

MOND suggests a modification of gravity if the test particle is moving with an
acceleration smaller than a critical value $a_{0}$. When the normal Newtonian
gravitational acceleration $g_{N}\ll a_{0}$ (i.e. the deep-MOND regime), the
gravity would be modified to (Sanders & McGaugh, 2002)

$g=\sqrt{g_{N}a_{0}}.$ (1)

When $g_{N}\gg a_{0}$ (i.e. the Newtonian regime), the gravity would restore
back to the Newtonian description: $g=g_{N}$. One can imagine that there is a
transition from $g=\sqrt{g_{N}a_{0}}$ to $g=g_{N}$ when $g_{N}$ is changing
from a very small value to a large value. In the MOND framework, the
transition is described by the interpolating function $\nu$ (Famaey & McGaugh,
2012). Therefore, the modification of gravity can be generally re-written as

$g=\nu(g_{N}/a_{0})g_{N},$ (2)

where $\nu(z)=1$ when $z\rightarrow\infty$ and $\nu(z)=\sqrt{1/z}$ when
$z\rightarrow 0$. Although there is no theoretical prediction of $\nu$ based
on MOND, there are some suggested families of functions which can satisfy the
required transition (Famaey & McGaugh, 2012):

$\nu(z)=\left[\frac{1+(1+4z^{-p})^{1/2}}{2}\right]^{1/p},$ (3)
$\nu(z)=[1-\exp(-z^{\delta/2})]^{-1/\delta}.$ (4)

Here, the parameters $p$ and $\delta$ are positive real numbers. For the
$p$-family, $p=1$ and $p=2$ are commonly known as the simple interpolating
function and the standard interpolating function respectively (Famaey &
McGaugh, 2012; Dutton et al., 2019). For the $\delta$-family, $\delta=1$ is
the well-known function which provides the best description of the radial
acceleration relation in galaxies (McGaugh, Lelli & Schombert, 2016; Dutton et
al., 2019). In any case, different possible values of $p$ and $\delta$ can
almost exhaustively represent all possible ways of transition between the
deep-MOND and Newtonian regimes (Dutton et al., 2019). Although most of the
previous studies took $p\geq 1$ and $\delta\geq 1$ (McGaugh, 2008; Dutton et
al., 2019), we generally allow that $p$ and $\delta$ can be any positive real
numbers first.

If dark matter does not exist, the gravity is fully contributed by the
baryonic matter. In a typical galaxy, the baryonic matter is dominated by the
bulge component and the disk component. Since we are concerning about the
outer region of a galaxy (i.e. large $R$), the bulge component can be well-
approximated by a point-mass function at the Galactic centre with the bulge
mass $M_{b}$. For the disk component, early observations have shown that the
disk surface brightness of many galaxies can be well-approximated by an
exponential function, including our Galaxy (Freeman, 1970; Jurić et al., 2008;
de Jong et al., 2010; McGaugh, 2016). Therefore, most of the past studies and
simulations have used an exponential function to model the surface mass
density of the disk component in our Galaxy (Misiriotis et al., 2006; Sofue,
2012; Bovy & Rix, 2013; Licquia & Newman, 2015):

$\Sigma=\Sigma_{0}e^{-R/a_{d}},$ (5)

where $a_{d}$ is the disk scale length and $\Sigma_{0}$ is the central surface
mass density. The total disk mass can be given by $M_{d}=2\pi
a_{d}^{2}\Sigma_{0}$. The Newtonian baryonic RC (without MOND transformation)
is given by (Freeman, 1970; Sofue, 2012)

$V_{\rm
bar}(x)=\sqrt{Ax^{2}\left[I_{0}(x)K_{0}(x)-I_{1}(x)K_{1}(x)\right]+\frac{B}{x}},$
(6)

where $A=2GM_{d}/a_{d}$, $B=GM_{b}/(2a_{d})$, $x=R/(2a_{d})$, $I_{n}$ and
$K_{n}$ are the modified Bessel functions of the $n^{\rm th}$ kind.

To make the analysis more comprehensive, apart from following the benchmark
exponential function to model the disk component, we will consider an extreme
case in which all of the baryonic disk mass is concentrated in $R<17$ kpc
(i.e. the concentrated baryonic disk model). This would represent a Keplerian
decrease in baryonic RC contribution and hence contribute to a more negative
RC slope. For this extreme scenario, the Newtonian baryonic RC without MOND
transformation is

$V_{\rm bar}(x)=\sqrt{\frac{A}{8x}+\frac{B}{x}}.$ (7)

Under the framework of MOND, we can transform the Newtonian baryonic RC to the
apparent RC $V(R)$ (i.e. the actual rotation curve observed) by using Eq. (2)
as

$V(x)=\sqrt{\nu\left(\frac{V_{\rm bar}^{2}(x)}{2a_{0}a_{d}x}\right)}V_{\rm
bar}(x).$ (8)

To understand the behavior of the RC, we take the derivative on $V(x)$ to get
the analytic forms of the RC gradient. For the $p$-family, by substituting Eq.
(3) into Eq. (8), we can get

$V(x)=2^{-1/2p}V_{\rm bar}\left(1+\sqrt{1+4y^{p}}\right)^{1/2p},$ (9)

where $y=2a_{0}a_{d}x/V_{\rm bar}^{2}$. Taking the derivative on both sides,
we can get the RC gradient for the $p$-family:

$\frac{dV}{dx}=\frac{(\sqrt{1+4y^{p}}+1+2y^{p})V^{\prime}_{\rm
bar}+2y^{p-1}a_{0}a_{d}V_{\rm
bar}^{-1}}{\sqrt{2^{1/p}(1+\sqrt{1+4y^{p}})^{2-1/p}(1+4y^{p})}}.$ (10)

For the $\delta$-family, we can write the apparent RC explicitly:

$V(x)=\sqrt{\left[1-\exp\left(-y^{-\delta/2}\right)\right]^{-1/\delta}}V_{\rm
bar}.$ (11)

Similarly, by taking the derivative on both sides, we get

$\frac{dV}{dx}=\left[1-\exp\left(-y^{-\delta/2}\right)\right]^{-1/2\delta}\left[V^{\prime}_{\rm
bar}+\frac{V_{\rm
bar}^{2}y^{1-\delta/2}}{4\exp(y^{-\delta/2})-4}\times\frac{V_{\rm
bar}-2xV^{\prime}_{\rm bar}}{2a_{0}a_{d}x^{2}}\right].$ (12)

Here, $V^{\prime}_{\rm bar}$ is the derivative on the Newtonian baryonic RC.
For the benchmark and the concentrated baryonic disk models, we have

$\displaystyle V^{\prime}_{\rm bar}$ $\displaystyle=$
$\displaystyle\frac{1}{2V_{\rm
bar}(x)}\left[2Ax\left(I_{0}(x)K_{0}(x)+xI_{1}(x)K_{0}(x)\right.\right.$ (13)
$\displaystyle\left.\left.-xI_{0}(x)K_{1}(x)\right)-\frac{B}{x^{2}}\right],$

and

$V^{\prime}_{\rm bar}=-\frac{1}{2V_{\rm
bar}}\left(\frac{A}{8x^{2}}+\frac{B}{x^{2}}\right)$ (14)

respectively. After grouping the appropriate terms in Eq. (12), we get the RC
gradient for the $\delta$-family:

$\frac{dV}{dx}=\frac{[\exp(y^{-\delta/2})-1-0.5y^{-\delta/2}]V^{\prime}_{\rm
bar}+0.25y^{-\delta/2}V_{\rm
bar}x^{-1}}{[1-\exp(-y^{-\delta/2})]^{1/2\delta}[\exp(y^{-\delta/2})-1]}.$
(15)

##  3 Data analysis

We test the MOND RC behavior by using the MWRC data. We focus on the outer
region $R\sim 20$ kpc of our Galaxy because MOND effect is significant when
$R$ is large. Recent accurate measurements of the MWRC at $R=17-27$ kpc can
give a rigorous examination of MOND. The most recent robust analysis of the
MWRC combining the data of APOGEE DR17, GAIA, 2MASS and WISE gives a large
decline in the RC gradient (Ou et al., 2023). However, since the systematic
uncertainties of the data are very large ($\sim 15$%) for $R>22.5$ kpc (Ou et
al., 2023), we will only analyze the data within the region $R=17.21-22.27$
kpc (systematic uncertainties $\sim 1-5$% only). Simple regression of the RC
data points gives $dV/dR=(-4.567\pm 0.532)$ km/s/kpc for $R=17.21-22.27$ kpc.
If we include the consideration of the small observational uncertainties for
each RC data point, the best-fit RC slope with $5\sigma$ margins is
$dV/dR=-5.07^{+2.51}_{-2.49}$ km/s/kpc (99.99994% C.L.) for $R=17.21-22.27$
kpc (see Fig. 1). The $5\sigma$ margins are calculated using the $\chi^{2}$
method. By examining different values of $dV/dR$ in the fits, the values of
the $5\sigma$ margins can be determined when the $\chi^{2}$ value is larger
than the $5\sigma$ critical value $\chi_{\rm crit}^{2}=48$ (degrees of freedom
= 10). Note that we did not include the systematic uncertainties of the RC
data points in determining the RC gradient. The actual effects of the
systematic uncertainties on the RC gradient depend on the involved factors. We
will discuss these effects in the discussion section.

From Eq. (10) and Eq. (15), we can see that the RC gradient depends on the
baryonic parameters ($M_{b}$, $M_{d}$ and $a_{d}$) and the MOND parameters
($a_{0}$, $p$ and $\delta$). In the followings, we take the standard value of
$a_{0}=1.2\times 10^{-8}$ cm/s2 to perform our analysis (McGaugh, Lelli &
Schombert, 2016). For the baryonic parameters, we first assume the fiducial
values adopted in Ou et al. (2023): $M_{b}=1.55\times 10^{10}M_{\odot}$,
$M_{d}=3.65\times 10^{10}M_{\odot}$ and $a_{d}=2.35$ kpc. These values are
determined by the direct baryonic observations (Misiriotis et al., 2006),
which are somewhat representative and good for performing the preliminary
analysis. We will also test a reasonable range of each parameter in the next
step.

In Fig. 2, we plot $dV/dR$ as a function of $p$ and $\delta$ for the
$p$-family and $\delta$-family interpolating functions respectively at
$R=19.71$ kpc. We have taken the data point at $R=19.71$ as the central
reference position because it is the median $R$ and the mean value of $R$
among the data points within the range $R=17.21-22.27$ kpc. Generally
speaking, our results do not depend on the choice of the central reference
position. We can see that larger values of $dV/dR$ (more positive or less
negative) would result when $p$ and $\delta$ are larger. For the benchmark
baryonic disk model, the $5\sigma$ allowed ranges of $p$ and $\delta$ are
$p=0.19-0.46$ and $\delta=0.17-0.43$ respectively. However, the Mercury
precession data have constrained $p\geq 1.22$ and $\delta\geq 0.33$ (Chan &
Lee, 2023). Therefore, a large range of $\delta$ and all values of $p$ are
ruled out based on the Mercury precession data and MWRC data. In particular,
the cases with $p=1$, $p=2$ and $\delta=1$ can give good agreements with the
data of galactic rotation curves (McGaugh, Lelli & Schombert, 2016; McGaugh,
2008; Dutton et al., 2019) and velocity dispersion profiles of elliptical
galaxies (Chae et al., 2020). However, these benchmark values are ruled out at
$5\sigma$ based on the MWRC slope with the fiducial values. Besides, even if
we consider the concentrated baryonic disk model, the values for $p\geq 1$ and
$\delta\geq 1$ are still ruled out at more than $5\sigma$ (see Fig. 2).

Apart from using the fiducial values of the baryonic parameters, we consider
conservative ranges of $M_{d}$ and $a_{d}$ for a more rigorous investigation.
Note that the room for varying the value of $M_{b}$ is very small because it
is mainly determined by the very inner RC data, which is almost independent of
MOND or any particular dark matter model. Therefore, we will still adopt
$M_{b}=1.55\times 10^{10}M_{\odot}$. Besides, previous studies have shown that
MOND favors small values of $a_{d}$ (Gerhard, 2002; McGaugh, 2008). The RC
data suggest $a_{d}\sim 2-4$ kpc and $M_{d}\sim(2.9-5.4)\times
10^{10}M_{\odot}$ under the MOND framework (McGaugh, 2008). These ranges are
generally consistent with the mass model predictions in many studies (Sofue,
2012; McGaugh, 2016; Ou et al., 2023). To perform a conservative test, we
allow wider ranges of $a_{d}$ and $M_{d}$: $a_{d}=1-10$ kpc and
$M_{d}=(1-10)\times 10^{10}M_{\odot}$.

In Fig. 3, we plot the RC gradient $dV/dR$ as a function of $a_{d}$ for
different $M_{d}$. Since most of the previous studies assume $p\geq 1$ and
$\delta\geq 1$ (Dutton et al., 2019; McGaugh, 2008), we only consider the
simplest form (the benchmark values) of $p=1$ and $\delta=1$ for both
interpolating families because larger values of $p$ and $\delta$ would make
the RC gradient larger. We find that almost all RC gradients are larger than
the $5\sigma$ upper bound of the observed RC gradients, except for the
$p$-family with the extreme parameter $M_{d}=10^{11}M_{\odot}$. This indicates
that a large parameter space of $a_{d}$ and $M_{d}$ in the conservative ranges
are ruled out at $5\sigma$ for $p\geq 1$ and $\delta\geq 1$.

##  4 Discussion

In this article, we present the RC gradient analysis for MOND using the MWRC
data at large $R$. Thanks to the high-quality measurements, the MWRC data have
very small error bars at large $R$, which give an excellent platform to test
modified gravity theories or constrain dark matter model. The MWRC data show a
very clear declining trend at large $R$ which potentially challenges MOND
because it predicts an almost flat RC at large $R$. The $5\sigma$ range of the
RC gradient is $dV/dR=-5.07^{+2.51}_{-2.49}$ km/s/kpc (without considering the
systematic uncertainties). On the other hand, by considering two major
families of the interpolating functions, we have derived the RC gradient
formulas explicitly for the MOND framework. We first follow the benchmark
baryonic disk density profile and use the fiducial values of $a_{d}$, $M_{d}$
and $M_{b}$ to calculate the RC gradient. There exist $5\sigma$ allowed ranges
of $p=0.19-0.46$ and $\delta=0.17-0.43$. However, when these are combined with
the Mercury precession constraints $p\geq 1.22$ and $\delta\geq 0.33$ (Chan &
Lee, 2023), only a narrow range of $\delta$ remains. This suggests that almost
all possible variations of MOND are ruled out. Even in the extreme case
(concentrated baryonic disk model), the benchmark MOND parameters $p\geq 1$
and $\delta\geq 1$ are still ruled out beyond $5\sigma$.

Besides, we also consider possible conservative ranges of $M_{d}=(1-10)\times
10^{10}M_{\odot}$ and $a_{d}=1-10$ kpc. For the benchmark cases $p=1$ and
$\delta=1$, almost all of the RC gradients calculated are above the $5\sigma$
upper bound of the observed RC gradient. As a larger $p$ or $\delta$ gives
less negative RC gradients, this gives a severe challenge to the MOND
phenomenology in our Galaxy. Nevertheless, we did not consider the systematic
uncertainties of the RC data points shown in Ou et al. (2023). Some of the
systematic uncertain factors (e.g. the solar position) would somewhat
systematically underestimate or overestimate the values of RC, without
affecting the value of the RC gradient. However, it is not sure how the other
factors discussed in Ou et al. (2023) would affect the RC gradient
calculation. If we assume 1% systematic uncertainties and combine with the
observational uncertainties, the $5\sigma$ range of the RC gradient would
enlarge to $dV/dR=-5.07^{+4.14}_{-3.77}$ km/s/kpc. However, if we focus on the
$2\sigma$ range instead, similar parameter space of the MOND parameters are
still ruled out at $2\sigma$ (95.45% C.L.) (see Fig. 3).

This is the first time we can rule out a large parameter space of galactic
MOND phenomenology at more than $5\sigma$. In fact, many previous studies,
like the SPARC analysis, have claimed that MOND works very well in galaxies
(McGaugh, 2020; Islam & Dutta, 2020). The radial acceleration relation shown
in SPARC gives an excellent agreement with MOND’s prediction (McGaugh, 2020;
McGaugh, Lelli & Schombert, 2016). However, the rotation curve data for the
galaxies outside our Local group have large uncertainties. Moreover, the
fittings also involve some unknown parameters like the mass-to-light ratio.
Therefore, there is plenty of room for MOND to fit with the data. In fact,
some later studies using better analysis tools have challenged the existence
of a universal acceleration scale $a_{0}$ (Rodrigues et al., 2018; Marra,
Rodrigues & de Almeida, 2020; Chan, Desai & Del Popolo, 2022) and the core-
cusp nature in galaxies (Eriksen, Frandsen & From, 2021) predicted by the MOND
phenomenology.

Fortunately, the very small error bars of the MWRC generated from the analysis
combining the data of APOGEE DR17, GAIA, 2MASS and WISE can give a robust test
for the galactic MOND phenomenology. We have tested two major families of
interpolating functions which can almost represent all possible transition
functions between the deep-MOND and Newtonian regimes. We have also considered
wide conservative ranges of $a_{d}$ and $M_{d}$. Our robust analysis gives a
severe challenge to the MOND phenomenology in our Galaxy, unless the effect of
the systematic uncertainties is larger than our expectation. The
phenomenological MOND might be just a rough approximation and it is definitely
not universal for all galaxies. Besides, the small uncertainties of the MWRC
generated can also be used to analyze the radial acceleration relation for our
Galaxy. This can further examine the MOND phenomenology and the alleged
universal acceleration scale shown in galaxies.

Figure 1: The black dots with error bars represent the data of the MWRC for
$R=17-23$ kpc (Ou et al., 2023). The red solid line indicates the best-fit RC
gradient $dV/dR=-5.07$ km/s/kpc. The red dotted lines indicate the $5\sigma$
upper and lower bounds of the best-fit RC gradients.

Figure 2: The black lines (assuming an exponential baryonic disk) and green
lines (assuming a small baryonic disk with radius $<17$ kpc) indicate the
functions of $dV/dR$ against $p$ (left) and $\delta$ (right). The red solid
line indicates the best-fit RC gradient and the shaded regions bounded by the
red dotted lines represent the parameter space ruled out at $5\sigma$ based on
the MWRC data. The blue shaded regions indicate the parameter space ruled out
at $1\sigma$ based on the Mercury precession data (Chan & Lee, 2023).

Figure 3: The colored lines (solid: $p$-family; dotted: $\delta$-family)
represent the MOND RC gradient $dV/dR$ against the disk scale length $a_{d}$
for different disk total mass $M_{d,10}$, where
$M_{d,10}=M_{d}/10^{10}M_{\odot}$. The red, black, and blue dashed lines
respectively indicate the best-fit RC gradient, its $5\sigma$ upper bound
(without considering systematic uncertainties), and its $2\sigma$ upper bound
(including 1% systematic uncertainties) from the MWRC data. The shaded region
is the region ruled out by the $5\sigma$ upper bound.

##  5 Acknowledgements

The work described in this paper was partially supported by a grant from the
Research Grants Council of the Hong Kong Special Administrative Region, China
(Project No. EdUHK 18300922).

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