The Rank-Ramsey Problem and the Log-Rank Conjecture

TL;DR

This paper introduces the Rank-Ramsey problem, explores its connection to the log-rank conjecture, and constructs families of graphs with specific rank and clique properties, advancing understanding in Ramsey theory and communication complexity.

Contribution

It systematically studies Rank-Ramsey graphs, constructs two families with polynomial separation in order and complement rank, and links these to the log-rank conjecture and matrix lifts.

Findings

Constructed two families of Rank-Ramsey graphs with polynomial separation.

Graphs with bounded clique number and small complement rank are demonstrated.

Established bounds on Rank-Ramsey numbers for low complement rank cases.

Abstract

A graph is called Rank-Ramsey if (i) Its clique number is small, and (ii) The adjacency matrix of its complement has small rank. We initiate a systematic study of such graphs. Our main motivation is that their constructions, as well as proofs of their non-existence, are intimately related to the famous log-rank conjecture from the field of communication complexity. These investigations also open interesting new avenues in Ramsey theory. We construct two families of Rank-Ramsey graphs exhibiting polynomial separation between order and complement rank. Graphs in the first family have bounded clique number (as low as $41$). These are subgraphs of certain strong products, whose building blocks are derived from triangle-free strongly-regular graphs. Graphs in the second family are obtained by applying Boolean functions to Erd\H{o}s-R\'enyi graphs. Their clique number is logarithmic, but their complement rank is far smaller than in the first family, about $\mathcal{O}(n^{2/3})$. A key component of this construction is our matrix-theoretic view of lifts. We also consider lower bounds on the Rank-Ramsey numbers, and determine them in the range where the complement rank is $5$ or less. We consider connections between said numbers and other graph parameters, and find that the two best known explicit constructions of triangle-free Ramsey graphs turn out to be far from Rank-Ramsey.