Threshold Ramsey multiplicity for odd cycles

TL;DR

This paper establishes a lower bound on the minimum number of monochromatic odd cycles in two-colorings of complete graphs at the Ramsey threshold, extending previous results for paths and even cycles.

Contribution

The authors prove that the threshold Ramsey multiplicity for odd cycles grows at least exponentially with the cycle length, matching bounds previously known for paths and even cycles.

Findings

Threshold Ramsey multiplicity for odd cycles is at least (ck)^k.

The result extends previous bounds from paths and even cycles to odd cycles.

Methods differ from prior work to establish the same exponential growth bound.

Abstract

The Ramsey number $r(H)$ of a graph $H$ is the minimum $n$ such that any two-coloring of the edges of the complete graph $K_n$ contains a monochromatic copy of $H$. The threshold Ramsey multiplicity $m(H)$ is then the minimum number of monochromatic copies of $H$ taken over all two-edge-colorings of $K_{r(H)}$. The study of this concept was first proposed by Harary and Prins almost fifty years ago. In a companion paper, the authors have shown that there is a positive constant $c$ such that the threshold Ramsey multiplicity for a path or even cycle with $k$ vertices is at least $(ck)^k$, which is tight up to the value of $c$. Here, using different methods, we show that the same result also holds for odd cycles with $k$ vertices.