Ramsey numbers of cycles in random graphs

TL;DR

This paper establishes sharp probabilistic bounds for the Ramsey numbers of cycles in random graphs, showing that with high probability, certain edge colorings contain monochromatic cycles under specified conditions.

Contribution

It improves existing bounds on Ramsey numbers of cycles in random graphs, providing sharper probabilistic thresholds and extending previous results.

Findings

High probability existence of monochromatic cycles in random graphs

Sharp bounds on the Ramsey number thresholds for cycles

Improved results over previous work by Letzter and Krivelevich et al.

Abstract

Let $R(C_n)$ be the Ramsey number of the cycle on $n$ vertices. We prove that, for some $C > 0$, with high probability every $2$-colouring of the edges of $G(N,p)$ has a monochromatic copy of $C_n$, as long as $N\geq R(C_n) + C/p$ and $p \geq C/n$. This is sharp up to the value of $C$ and it improves results of Letzter and of Krivelevich, Kronenberg and Mond.