Ramsey numbers and the Zarankiewicz problem

TL;DR

This paper links Ramsey numbers of certain graphs to a variant of the Zarankiewicz problem, providing new bounds for specific cycles and suggesting a method to approximate these numbers for larger odd cycles.

Contribution

It establishes a general connection between Ramsey numbers and Zarankiewicz-type problems, leading to new bounds for cycles and a framework for approximating Ramsey numbers of odd cycles.

Findings

New lower bounds for r(C_5,t) and r(C_7,t)

Connection between Ramsey numbers and Zarankiewicz problem

Potential to approximate r(C_{2 extstyle ext{l}+1}, t) for larger extstyle ext{l}

Abstract

Building on recent work of Mattheus and Verstra\"ete, we establish a general connection between Ramsey numbers of the form $r(F,t)$ for $F$ a fixed graph and a variant of the Zarankiewicz problem asking for the maximum number of 1s in an $m$ by $n$ $0/1$-matrix that does not have any matrix from a fixed finite family $\mathcal{L}(F)$ derived from $F$ as a submatrix. As an application, we give new lower bounds for the Ramsey numbers $r(C_5,t)$ and $r(C_7,t)$, namely, $r(C_5,t) = \tilde\Omega(t^{\frac{10}{7}})$ and $r(C_7,t) = \tilde\Omega(t^{\frac{5}{4}})$. We also show how the truth of a plausible conjecture about Zarankiewicz numbers would allow an approximate determination of $r(C_{2\ell+1}, t)$ for any fixed integer $\ell \geq 2$.