Threshold Ramsey multiplicity for paths and even cycles

TL;DR

This paper investigates the minimum number of monochromatic paths and cycles in two-colorings of complete graphs at the Ramsey threshold, establishing tight exponential lower bounds for these quantities.

Contribution

It proves that the threshold Ramsey multiplicity for paths and even cycles grows exponentially with the number of vertices, providing tight bounds and addressing a longstanding problem.

Findings

Threshold Ramsey multiplicity for paths and even cycles is at least (ck)^k.

The bound is tight up to a constant factor.

Similar results are obtained for odd cycles in a companion paper.

Abstract

The Ramsey number $r(H)$ of a graph $H$ is the minimum integer $n$ such that any two-coloring of the edges of the complete graph $K_n$ contains a monochromatic copy of $H$. While this definition only asks for a single monochromatic copy of $H$, it is often the case that every two-edge-coloring of the complete graph on $r(H)$ vertices contains many monochromatic copies of $H$. The minimum number of such copies over all two-colorings of $K_{r(H)}$ will be referred to as the threshold Ramsey multiplicity of $H$. Addressing a problem of Harary and Prins, who were the first to systematically study this quantity, we show that there is a positive constant $c$ such that the threshold Ramsey multiplicity of a path or an even cycle on $k$ vertices is at least $(ck)^k$. This bound is tight up to the constant $c$. We prove a similar result for odd cycles in a companion paper.