Schur's colouring theorem for non-commuting pairs

TL;DR

This paper investigates the relationship between the probability of elements commuting in finite non-Abelian groups and the minimal coloring size needed to guarantee monochromatic quadruples with non-commuting pairs.

Contribution

It establishes a precise equivalence between the decay of commutativity probability and the unboundedness of the coloring number for such groups.

Findings

c(G) tends to 0 iff k(G) tends to infinity

Provides a new link between algebraic structure and combinatorial coloring

Characterizes the asymptotic behavior of non-Abelian groups

Abstract

For G a finite non-Abelian group we write c(G) for the probability that two randomly chosen elements commute and k(G) for the largest integer such that any k(G)-colouring of G is guaranteed to contain a monochromatic quadruple (x,y,xy,yx) with xy not equal to yx. We show that c(G) tends to 0 if and only if k(G) tends to infinity.

Equations

$$c(G):=\frac{1}{|G{|}^2}\sum _{\_}x,y\in G{1}_{\_}[xy=yx];$$

$$S(\mathcal{A}):=\frac{1}{|G{|}^3}\sum _{\_}x,y,z\in G{1}_{\_}\mathcal{A}(x,y)1_{\_}\mathcal{A}(zx,y)1_{\_}\mathcal{A}(x,yz)⩾f_{\_}\Delta (\alpha ).$$

$$\frac{1}{3}{f}_{\_}\Delta ({\alpha }_{\_}1\cdots {\alpha }_{\_}r)⩾{\alpha }_{\_}r+1+\cdots +{\alpha }_{\_}k,$$

$${\alpha }_{\_}i+1>\frac{1}{3k}{f}_{\_}\Delta ({\alpha }_{\_}1\cdots {\alpha }_{\_}i)\text{ for all }1⩽i⩽r-1.$$

$${\mathcal{A}}_{\_}i:=\{(x,y)\in G^2:x{s}_{\_}iy\in A_{\_}i\}\text{ for }1⩽i⩽r.$$

$${\mathbb{E}}_{\_}{s}_{\_}i\in G{1}_{\_}{\mathcal{A}}_{\_}i(x,y)={\alpha }_{\_}i\text{ for all }x,y\in G\text{ and }1⩽i⩽r,$$

$${\mathbb{E}}_{\_}s\in G^r\left|\bigcap _{\_}{i=1}^r{\mathcal{A}}_{\_}i\right|=\sum _{\_}x,y\in G{\mathbb{E}}_{\_}s\in G^r\prod _{\_}{i=1}^r{1}_{\_}{\mathcal{A}}_{\_}i(x,y)={\alpha }_{\_}1\cdots {\alpha }_{\_}r|G{|}^2.$$

$${\mathbb{E}}_{\_}x,y,z\in G{1}_{\_}\mathcal{A}(x,y)1_{\_}\mathcal{A}(zx,y)1_{\_}\mathcal{A}(x,yz)=S(\mathcal{A})⩾f_{\_}\Delta ({\alpha }_{\_}1\cdots {\alpha }_{\_}r);$$

$$Z:=\left\{z\in G:{\mathbb{E}}_{\_}x,y\in G{1}_{\_}\mathcal{A}(x,y)1_{\_}\mathcal{A}(zx,y)1_{\_}\mathcal{A}(x,yz)⩾\frac{1}{3}f({\alpha }_{\_}1\cdots {\alpha }_{\_}r)\right\}.$$

$$\mathbb{P}(Z)+\frac{1}{3}{f}_{\_}\Delta ({\alpha }_{\_}1\cdots {\alpha }_{\_}r)$$

$$\mathbb{P}(Z\setminus (A_{\_}r+1\cup \cdots \cup A_{\_}k))$$

$$\mathbb{P}((Z\setminus (A_{\_}r+1\cup \cdots \cup A_{\_}k))\cap A_{\_}i)⩾\frac{1}{3r}{f}_{\_}\Delta ({\alpha }_{\_}1\cdots {\alpha }_{\_}r).$$

$${\mathbb{E}}_{\_}x,y{1}_{\_}{\mathcal{A}}_{\_}i(x,y)1_{\_}{\mathcal{A}}_{\_}i(zx,y)1_{\_}{\mathcal{A}}_{\_}i(x,yz)⩾{\mathbb{E}}_{\_}x,y{1}_{\_}\mathcal{A}(x,y)1_{\_}\mathcal{A}(zx,y)1_{\_}\mathcal{A}(x,yz)⩾\frac{1}{3}{f}_{\_}\Delta ({\alpha }_{\_}1\cdots {\alpha }_{\_}r)$$

$$\frac{1}{3}{f}_{\_}\Delta ({\alpha }_{\_}1\cdots {\alpha }_{\_}r)|G{|}^2\cdot \frac{1}{3r}{f}_{\_}\Delta ({\alpha }_{\_}1\cdots {\alpha }_{\_}r)|G|$$

$$z\in A_{\_}i,x{s}_{\_}iy\in A_{\_}i,zx{s}_{\_}iy\in A_{\_}i,\text{ and }x{s}_{\_}iyz\in A_{\_}i.$$

$$\frac{1}{9r}{f}_{\_}\Delta ({\alpha }_{\_}1\cdots {\alpha }_{\_}r{)}^2|G{|}^2⩾(g({\alpha }_{\_}r){)}^2|G{|}^2$$

$$X:=\{x\in G:|x^G|⩽{\eta }^{-1}\}.$$

$$ϵ|G{|}^2⩽|G{|}^2\mathbb{P}(xy=yx)=\sum _{\_}x|C_{\_}G(x)|=|G|\sum _{\_}x\frac{1}{|x^G|}⩽\sum _{\_}x\in X|G|+\sum _{\_}x\notin X\eta |G|.$$

$$|\stackrel{s\text{ times}}{\overbrace{X\cdots X}}|⩾\left(1+(1-\nu )(s-1)\right)|X|.$$

$$|\stackrel{s+1\text{ times}}{\overbrace{X\cdots X}}|⩾|\stackrel{s\text{ times}}{\overbrace{X\cdots X}}|+|X|-|H|\text{ and }H\subset \stackrel{s+1\text{ times}}{\overbrace{X\cdots X}}.$$

$$\left(1+(1-\nu )s\right)|X|>|\stackrel{s+1\text{ times}}{\overbrace{X\cdots X}}|⩾\left(1+(1-\nu )(s-1)\right)|X|+|X|-|H|,$$

$$K◃G\text{ and }|G/K|⩽|\mathrm{Sym}(G/H)|⩽|G/H|!.$$

$$|x^G|⩽{\eta }^{-(s+1)}⩽R:=\lfloor {\eta }^{-\frac{{\kappa }^{-1}+1-2\nu }{1-\nu }}\rfloor \text{ for all }x\in X^{s+1},$$

$$\mathcal{S}:=\{\{x\in K:{\varphi }_{\_}{x}^G(x)=i\}:1⩽i⩽R\}\text{ and }\mathcal{C}:=((G/K)\setminus \{K\})\cup \mathcal{S},$$

$$|\mathcal{C}|⩽|G/K|-1+|\mathcal{S}|$$

$$⩽\exp \left(\max \left\{{\nu }^{-1}{\kappa }^{-1}\log {\nu }^{-1}{\kappa }^{-1},\frac{{\kappa }^{-1}+1-2\nu }{1-\nu }\log {\eta }^{-1}\right\}+O(1)\right).$$

Full text

1. Introduction

Our starting point is Schur’s theorem [Sch16, Hilffssatz], the proof of which adapts to give the following.

Theorem 1.1**.**

Suppose that $G$ is a finite group and $\mathcal{C}$ is a cover of $G$ of size $k$. Then there is a set $A\in\mathcal{C}$ with at least $c_{\_}k|G|^{2}$ triples $(x,y,xy)\in A^{3}$ where $c_{\_}k$ is a constant depending only on $k$.

The proof is a routine adaptation, but we shall not give it as the result as stated also follows from our next theorem.

If $G$ is non-Abelian then we might like to ask for quadruples $(x,y,xy,yx)\in A^{4}$ instead of triples and establishing the following such result (which we do in §2) is the main purpose of the paper.

Theorem 1.2**.**

Suppose that $G$ is a finite group and $\mathcal{C}$ is a cover of $G$ of size $k$. Then there is a set $A\in\mathcal{C}$ with $c_{\_}k|G|^{2}$ quadruples $(x,y,xy,yx)\in A^{4}$ where $c_{\_}k$ is a constant depending only on $k$.

When $G$ is non-Abelian we should like to ensure that at least one of the quadruples found in Theorem 1.2 has $xy\neq yx$, and to this end we define the commuting probability of a finite group $G$ to be

[TABLE]

in words it is the probability that a pair $(x,y)\in G^{2}$ chosen uniformly at random has $xy=yx$. There are many nice results about the commuting probability – see the introduction to [Heg13] for details – and it is an instructive exercise (see [Gus73]) to check that if $c(G)<1$ then $c(G)\leqslant\frac{5}{8}$, so that if a group is non-Abelian there are ‘many’ pairs that do not commute. Despite this we have the following which we prove in §3.

Proposition 1.3**.**

Suppose that $G$ is a finite group and $c(G)\geqslant\epsilon$. Then there is a cover $\mathcal{C}$ of $G$ of size $\exp((2+o_{\_}{\epsilon\rightarrow 0}(1))\epsilon^{-1}\log\epsilon^{-1})$ such that if $A\in\mathcal{C}$ and $(x,y,xy,yx)\in A^{4}$ then $xy=yx$.

If $G$ is non-Abelian we write $k(G)$ for the non-commuting Schur number of $G$, that is the largest natural number such that for any cover $\mathcal{C}$ of $G$ of size $k(G)$ there is some $A\in\mathcal{C}$ and $(x,y,xy,yx)\in A^{4}$ with $xy\neq yx$. (Note that since $G$ is assumed non-Abelian we certainly have $k(G)\geqslant 1$.)

The number $k(G)$ has been studied for a range of specific groups by McCutcheon in [McC] and we direct the interested reader there for examples and further questions.

Theorem 1.4**.**

Suppose that $(G_{\_}n)_{\_}n$ is a sequence of non-Abelian groups. Then $c(G_{\_}n)\rightarrow 0$ if and only if $k(G_{\_}n)\rightarrow\infty$.

Proof.

The right to left implication follows immediately from Proposition 1.3. We can assume that $c_{\_}k$ is monotonically decreasing. Suppose that $c(G_{\_}n)\rightarrow 0$ and there is a $k_{\_}0$ and an infinite set $S$ of naturals such that $k(G_{\_}n)<k_{\_}0$ for all $n\in S$. Let $n\in S$ be such that $c(G_{\_}n)<c_{\_}{k_{\_}0}$ which can be done since $c(G_{\_}n)\rightarrow 0$ and $c_{\_}{k_{\_}0}>0$.

Since $k(G_{\_}n)<k_{\_}0$ there is a cover $\mathcal{C}$ of $G_{\_}n$ of size $k_{\_}0$ such that if $A\in\mathcal{C}$ and $(x,y,xy,yx)\in A^{4}$ then $xy=yx$. By Theorem 1.2 there is $A\in\mathcal{C}$ such that $(x,y,xy,yx)\in A^{4}$ for at least $c_{\_}{k_{\_}0}|G_{\_}n|^{2}$ quadruples. But then by design $xy=yx$ for all these pairs and so $c(G_{\_}n)\geqslant c_{\_}{k_{\_}0}$, a contradiction which proves the result. ∎

Before closing the section we need to acknowledge our debt to previous work. In [McC12] McCutcheon proves that $k(S_{\_}n)\rightarrow\infty$ as $n\rightarrow\infty$. A short calculation shows that $c(S_{\_}n)\rightarrow 0$ as $n\rightarrow\infty$, and the possibility of showing that $k(G_{\_}n)\rightarrow\infty$ as $c(G_{\_}n)\rightarrow 0$ is identified by Bergelson and Tao in the remarks after [BT14, Theorem 11]. Earlier, in [BT14, Footnote 4], they also observe the significance of Neumann’s work [Neu89] which is the main idea behind the proof of Proposition 1.3.

Write $D(G)$ for the smallest dimension of a non-trivial unitary representation of $G$.111This is called the quasirandomness of $G$ in [BT14, Definition 1] following the work of Gowers [Gow08] In [BT14, Corollary 8] the authors show that $k(G_{\_}n)\rightarrow\infty$ as $D(G_{\_}n)\rightarrow\infty$, and in fact go further proving a density result. For general finite groups there can be no density result; we refer the reader to the discussion after [BT14, Theorem 11] for more details.

2. Proof of Theorem 1.2

The proof of Theorem 1.2 is inspired by an attempt to translate the proof of [BM98, Theorem 3.4] into a combinatorial setting. There the authors use a recurrence theorem [BMZ97, Theorem 5.2]; in its place we use a version of the Ajtai-Szemerédi Corners Theorem [AS74] for finite groups. This was proved by Solymosi [Sol13, Theorem 2.1] using the triangle removal lemma.

Theorem 2.1**.**

There is a function $f_{\_}\Delta:(0,1]\rightarrow(0,1]$ such that if $G$ is a finite group and $\mathcal{A}\subset G^{2}$ has size at least $\alpha|G|^{2}$ then

[TABLE]

Proof.

Following the proof of [Sol13, Theorem 2.1], form a tripartite graph with three copies of $G$ as the vertex sets (call them $V_{\_}1,V_{\_}2,V_{\_}3$) and joining $(x,y)\in V_{\_}1\times V_{\_}2$ if and only if $(x,y)\in\mathcal{A}$; $(y,w)\in V_{\_}2\times V_{\_}3$ if and only if $(y^{-1}w,y)\in\mathcal{A}$; and $(x,w)\in V_{\_}1\times V_{\_}3$ if and only if $(x,wx^{-1})\in\mathcal{A}$. The map $G^{3}\rightarrow G^{3};(x,y,w)\mapsto(x,y,y^{-1}wx^{-1})$ is a bijection and $(x,y,w)$ is a triangle in this graph if and only if $(x,y),(zx,y),(x,yz)\in\mathcal{A}$ where $z=y^{-1}wx^{-1}$.

It follows from [Tao06, Theorem 1.1] that one can remove at most $3\cdot o_{\_}{S(\mathcal{A})\rightarrow 0}(|G|^{2})=o_{\_}{S(\mathcal{A})\rightarrow 0}(|G|^{2})$ elements from $\mathcal{A}$ to make the graph triangle-free. On the other hand if $(x,y)\in\mathcal{A}$ then $(x,y,xy)$ is a triangle in the above graph, hence we must have removed all elements from $\mathcal{A}$ and $\alpha|G|^{2}\leqslant o_{\_}{S(\mathcal{A})\rightarrow 0}(|G|^{2})$ from which the result follows. ∎

There are a number of subtleties around the extent to which one can replace, say, $(zx,y)$ with $(xz,y)$, and we refer the reader to the papers of Solymosi [Sol13] and Austin [Aus16] for some discussion.

We take the convention, as we can, that the function $f_{\_}\Delta$ is monotonically increasing and $f_{\_}\Delta(x)\leqslant x$ for all $x\in(0,1]$. Even with Fox’s work [Fox11] we only have $f_{\_}\Delta(\alpha)^{-1}\leqslant T(O(\log\alpha^{-1}))$ in general. However, when $G$ is Abelian much better bounds are known as a result of the beautiful arguments of Shkredov [Shk06b, Shk06a, Shk09]. It seems likely these could be adapted to give a bound with a tower of bounded height if the Fourier analysis is adapted to the non-Abelian setting in the same way as it is for Roth’s theorem in [San17]. Doing so would give a quantitative version of [BT14, Theorem 10] (see [BT14, Remark 44]), but the improvement to Theorem 1.2 would only be to replace a wowzer-type function with a tower as we shall see shortly.

We shall prove the following proposition from which Theorem 1.2 follows immediately on inserting the bound for $f_{\_}\Delta$ given by Theorem 2.1.

Proposition 2.2**.**

Suppose $G$ is a finite group and $\mathcal{C}$ is a cover of $G$ of size $k$. Then there is a set $A\in\mathcal{C}$ with $(g^{(k+1)}(1))^{2}|G|^{2}$ quadruples $(x,y,xy,yx)\in A^{4}$ where $g^{(k+1)}$ is the $(k+1)$-fold composition of $g$ with itself, and $g:(0,1]\rightarrow(0,1];\alpha\mapsto(3k)^{-1}f_{\_}\Delta(\alpha^{k})$.

Proof.

Write $A_{\_}1,\dots,A_{\_}k$ for the sets in $\mathcal{C}$ ordered so that they have density $\alpha_{\_}1\geqslant\dots\geqslant\alpha_{\_}k$ respectively; since $\mathcal{C}$ is a cover we have $\alpha_{\_}1\geqslant\frac{1}{k}$. Let $r\in\{1,\dots,k\}$ be minimal such that

[TABLE]

which is possible since the sum on the right is empty and so [math] when $r=k$. From minimality and the order of the $\alpha_{\_}i$s we have

[TABLE]

The function $f_{\_}\Delta$ is monotonically increasing and $f_{\_}\Delta(x)\leqslant x$ for all $x\in(0,1]$ so it follows from the above that $\alpha_{\_}r\geqslant g^{(r)}(1)\geqslant g^{(k)}(1)$.

Now, suppose that $s_{\_}1,\dots,s_{\_}r\in G$ and write

[TABLE]

Then

[TABLE]

and so

[TABLE]

By averaging we can pick some $s\in G^{r}$ such that $\mathcal{A}:=\bigcap_{\_}{i=1}^{r}{\mathcal{A}_{\_}i}$ has $|\mathcal{A}|\geqslant\alpha_{\_}1\cdots\alpha_{\_}r|G|^{2}$.

By the definition of $f_{\_}\Delta$ (from Theorem 2.1) we have

[TABLE]

write

[TABLE]

Then

[TABLE]

and hence $\mathbb{P}(Z)\geqslant\frac{2}{3}f_{\_}\Delta(\alpha_{\_}1\cdots\alpha_{\_}r)$. But then

[TABLE]

by (2.1). Since $\bigcup_{\_}{i=1}^{k}{A_{\_}i}=G$ we conclude that there is some $1\leqslant i\leqslant k$ such that

[TABLE]

If course $(Z\setminus(A_{\_}{r+1}\cup\dots\cup A_{\_}{k}))\cap A_{\_}j=\emptyset$ for $r<j\leqslant k$ and so we may assume $i\leqslant r$.

Write $Z^{\prime}:=(Z\setminus(A_{\_}{r+1}\cup\dots\cup A_{\_}{k}))\cap A_{\_}i$. Since $Z^{\prime}\subset Z$ we have

[TABLE]

for all $z\in Z^{\prime}$. On the other hand every $z\in Z^{\prime}$ has $z\in A_{\_}i$ and so we conclude that there are at least

[TABLE]

triples $(x,y,z)\in G^{3}$ such that

[TABLE]

The map $(x,y,z)\mapsto(xs_{\_}iy,z)$ has all fibres of size $|G|$ and so there are at least

[TABLE]

pairs $(a,b)\in G^{2}$ such that $a,b,ab,ba\in A_{\_}i$. This gives the result. ∎

3. Proof of Proposition 1.3

The key idea comes from Neumann’s theorem [Neu89, Theorem 1] which is already identified in [BT14, Footnote 4]. Neumann’s theorem describes the structure of groups $G$ for which $c(G)\geqslant\epsilon$ – they are the groups containing normal subgroups $K\leqslant H\leqslant G$ such that $K$ and $G/H$ have size $O_{\_}{\epsilon}(1)$ and $H/K$ is Abelian. Neumann’s theorem was further developed in [Ebe15, Theorem 2.4], but both arguments provide a more detailed structure than we require.

We have made some effort to control the exponent; results such as [Ebe15, Lemma 2.1] or [Ols76, Theorem 2.2] could be used in place of Kemperman’s Theorem in what follows at the possible expense of the $2$ becoming slightly larger. Moving the $2+o_{\_}{\epsilon\rightarrow 0}(1)$ below $1$ would require a slightly different approach as we normalise a subgroup of index around $\epsilon^{-1}$ at a certain point which costs us a term of size $\epsilon^{-1}!$.

Proposition** (Proposition 1.3).**

Suppose that $G$ is a finite group and $c(G)\geqslant\epsilon$. Then there is a cover $\mathcal{C}$ of $G$ of size $\exp((2+o_{\_}{\epsilon\rightarrow 0}(1))\epsilon^{-1}\log\epsilon^{-1})$ such that if $A\in\mathcal{C}$ and $(x,y,xy,yx)\in A^{4}$ then $xy=yx$.

Proof.

We work with the conjugation action of $G$ on itself (i.e. $(g,x)\mapsto g^{-1}xg$), and write $x^{G}$ for the conjugacy class of $x$ (the orbit of $x$ under this action), and $C_{\_}G(x)$ for the centre of $x$ in $G$ (the stabiliser of $x$ under this action).

Let $\eta,\nu\in(0,1]$ be parameters (we shall take $\nu=\frac{1}{2}$ and $\eta=\epsilon/\log\epsilon^{-1})$) to be optimised later and put

[TABLE]

Then

[TABLE]

Writing $\kappa:=|X|/|G|$ we can rearrange the above to see that $\kappa\geqslant(\epsilon-\eta)/(1-\eta)$.

Suppose that $s\in\mathbb{N}$ is maximal such that

[TABLE]

Certainly there is some $s\in\mathbb{N}$ since the inequality certainly holds for $s=1$, and there is a maximal such with $s\leqslant\frac{\kappa^{-1}-\nu}{1-\nu}$ since $|X|\geqslant\kappa|G|$.

Since $1_{\_}G^{G}=\{1_{\_}G\}$ we have $1_{\_}G\in X$ and $1_{\_}G\in X\cdots X$ for any $s$-fold product. By Kemperman’s Theorem [Kem56, Theorem 5] (also recorded on [Ols84, p111], and which despite the additive notation does not assume commutativity) it follows that there is some $H\leqslant G$ such that

[TABLE]

By maximality of $s$ we know that

[TABLE]

so that $|H|>\nu|X|$, and so $|G/H|<\nu^{-1}\kappa^{-1}$.

Let $K$ be the kernel of the action of left multiplication by $G$ on $G/H$ i.e. $K:=\{x\in G:xgH=gH\text{ for all }g\in G\}$. The action induces a homomorphism from $G$ to $\mathop{\rm Sym}\nolimits(G/H)$ so that by the First Isomorphism Theorem

[TABLE]

Each $x\in H$ (and hence each $x\in K$ since $xH=H$ for such $x$) can be written as a product of $s+1$ elements of $X$. Moreover, the function $x\mapsto|x^{G}|$ is sub-multiplicative i.e. $|(xy)^{G}|\leqslant|x^{G}||y^{G}|$ and so it follows that

[TABLE]

and in particular for all $x\in K$. Thus for each $x\in K$ there is an injection $\phi_{\_}{x^{G}}:x^{G}\rightarrow\{1,\dots,R\}$. With this notation we can define our covering: let

[TABLE]

so that $\mathcal{S}$ is a cover of $K$ and $\mathcal{C}$ is a cover of $G$. Now

[TABLE]

Optimise this by taking $\nu=\frac{1}{2}$ and $\eta=\epsilon/\log\epsilon^{-1}$ as mentioned before so that $\kappa\geqslant\epsilon(1-o_{\_}{\epsilon\rightarrow 0}(1))$ and $\log\eta^{-1}=(1+o_{\_}{\epsilon\rightarrow 0}(1))\log\epsilon^{-1}$.

Suppose that $A\in\mathcal{C}$ and $x,y,xy,yx\in A$. If $A\in(G/K)\setminus\{K\}$ then $xK=yK=xyK=yxK=A$. Since $K\triangleleft G$ we have $xK=xyK=(xK)(yK)$ and so $yK=K$ which is a contradiction. It follows that $A\in\mathcal{S}$ and hence $x,y,xy,yx\in K$. We conclude $\phi_{\_}{(xy)^{G}}(xy)=\phi_{\_}{(yx)^{G}}(yx)$ but $xy=y^{-1}(yx)y$ and so $(xy)^{G}=(yx)^{G}$. Since $\phi_{\_}{(xy)^{G}}$ is an injection we conclude that $xy=yx$ as required.

The result is proved. ∎