Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Rank-Ramsey Problem and the Log-Rank Conjecture

Gal Beniamini, Nati Linial, Adi Shraibman

TL;DR

This work constructs two families of Rank-Ramsey graphs exhibiting polynomial separation between order and complement rank, and considers lower bounds on the Rank-Ramsey numbers, and considers connections between said numbers and other graph parameters.

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 Erdős-Rényi 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.

The Rank-Ramsey Problem and the Log-Rank Conjecture

TL;DR

This work constructs two families of Rank-Ramsey graphs exhibiting polynomial separation between order and complement rank, and considers lower bounds on the Rank-Ramsey numbers, and considers connections between said numbers and other graph parameters.

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 ). 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 Erdős-Rényi graphs. Their clique number is logarithmic, but their complement rank is far smaller than in the first family, about . 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 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.
Paper Structure (39 sections, 37 theorems, 101 equations, 1 figure)

This paper contains 39 sections, 37 theorems, 101 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Our Results.
  3. Paper Organization
  4. Log-Rank.
  5. Constructions.
  6. Analysis and Bounds.
  7. Preliminaries
  8. Graphs.
  9. Boolean Functions.
  10. Matrices.
  11. Communication Complexity.
  12. Matrix Representation of Lifts
  13. Rank-Ramsey Graphs
  14. Rank-Ramsey Numbers
  15. Warmup: Simple Constructions of Rank-Ramsey Graphs
  16. ...and 24 more sections

Key Result

Theorem 1

The following bounds hold:

Figures (1)

  • Figure 1: Illustration of the Kronecker product $\mathcal{C} \otimes K_3$, where $\mathcal{C}$ is the Clebsch graph. Similarly to blowup, vertices are replaced by anticliques, and edges by bipartite graphs. The key difference is that here edges are replaced by complete bipartite graphs minus the identity matching (so the graph is twin-free).

Theorems & Definitions (85)

  • Definition
  • Theorem 1
  • Theorem 2
  • Definition
  • Theorem 3
  • Theorem 4
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • proof
  • ...and 75 more