On the clique number of random Cayley graphs and related topics

TL;DR

The paper analyzes the clique number of random Cayley graphs over finite groups, establishing a general bound $O(\log N\log\log N)$ with high probability and showing this is tight for groups like $\mathbb{F}_2^n$. It develops a broad combinatorial framework—random entangled graphs and color-based subset counting—to derive powerful bounds on the number of small-color subsets and the additive-doubling structure of sets. These tools enable progress on Alon's Ramsey Cayley conjecture, proving it for broad classes including abelian groups of almost all orders and vector spaces with certain characteristics, and yield self-complementary Ramsey Cayley graphs with optimal $\,(2+o(1))\log N$ clique/independence numbers. The results connect graph Ramsey theory, additive combinatorics (Freiman dimension, doubling), and information-theoretic questions, with implications for related random-graph models and extractors.

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.

Theorems & Definitions (81)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Corollary 1.6
• Corollary 1.7
• Theorem 1.8
• Conjecture 1.9: Alon
• Theorem 1.10