On the clique number of random Cayley graphs and related topics

TL;DR

This paper establishes tight bounds on the clique number of random Cayley graphs across various groups, introduces a combinatorial proof technique, and confirms a conjecture for abelian groups, with implications for Ramsey theory and information theory.

Contribution

It provides the first tight bounds on clique numbers of random Cayley graphs, introduces a purely combinatorial proof method, and verifies a conjecture for abelian groups.

Findings

Clique number of random Cayley graphs is O(log N log log N) with high probability.

The bound is optimal for certain groups like _2^n.

Existence of self-complementary Cayley graphs with small clique and independence numbers for specific vector spaces.

Abstract

We prove that a random Cayley graph on a group of order $N$ has clique number $O(\log N \log \log N)$ with high probability. This bound is best possible up to the constant factor for certain groups, including~$\mathbb{F}_2^n$, and improves the longstanding upper bound of $O(\log^2 N)$ due to Alon. Our proof does not make use of the underlying group structure and is purely combinatorial, with the key result being an essentially best possible upper bound for the number of subsets of given order that contain at most a given number of colors in a properly edge-colored complete graph. As a further application of this result, we study a conjecture of Alon stating that every group of order $N$ has a Cayley graph whose clique number and independence number are both $O(\log N)$, proving the conjecture for all abelian groups of order $N$ for almost all $N$. For finite vector spaces of order $N$ with characteristic congruent to $1 \pmod 4$, we prove the existence of a self-complementary Cayley graph on the vector space whose clique number and independence number are both at most $(2+o(1))\log N$. This matches the lower bound for Ramsey numbers coming from random graphs and solves, in a strong form, a problem of Alon and Orlitsky motivated by information theory.