Number of sets with small sumset and the clique number of random Cayley graphs

TL;DR

This paper extends Green's results on the size of the largest clique or independent set in Cayley graphs to all finite abelian groups, showing bounds depend on the number of prime divisors.

Contribution

It generalizes Green's bounds from vector spaces over finite fields to all finite abelian groups, incorporating the number of prime divisors into the size estimates.

Findings

Cayley graphs can be constructed with small maximum clique and independent set sizes.

Bounds depend on the number of prime divisors of the group order.

Results unify and extend previous bounds for specific group types.

Abstract

Let $G$ be a finite abelian group of order $n$. For any subset $B$ of $G$ with $B=-B$, the Cayley graph $G_B$ is a graph on vertex set $G$ in which $ij$ is an edge if and only if $i-j\in B.$ It was shown by Ben Green that when $G$ is a vector space over a finite field $Z/pZ$, then there is a Cayley graph containing neither a complete subgraph nor an independent set of size more than $clog nloglog n,$ where $c$ is an absolute constant. In this article we observe that a modification of his arguments shows that for an arbitrary finite abelian group of order $n$, there is a Cayley graph containing neither a complete subgraph nor an independent set of size more than $c(omega^3(n)log omega(n) +log nloglog n)$, where $c$ is an absolute constant and $omega(n)$ denotes the number of distinct prime divisors of $n$.