Sharp Ramsey thresholds for large books

TL;DR

This paper establishes sharp thresholds for the Ramsey property of large book graphs in random graphs, extending known results on book Ramsey numbers and providing precise probabilistic thresholds.

Contribution

It introduces the sharp Ramsey threshold for large book graphs in random graphs, extending previous work on book Ramsey numbers and thresholds.

Findings

Identifies the threshold probability for the Ramsey property of large books in G(N,p).

Shows the threshold is sharp and depends on parameters c, k, and n.

Extends previous bounds on book Ramsey numbers to probabilistic thresholds.

Abstract

For graphs $G$ and $H$, let $G\to H$ signify that any red/blue edge coloring of $G$ contains a monochromatic $H$. Let $G(N,p)$ be the random graph of order $N$ and edge probability $p$. The Ramsey thresholds for fixed graphs have received most attention. In this paper, we consider the Ramsey thresholds in another angle. In particular, we will consider the sharp Ramsey threshold for the large book graph $B_n^{(k)}$, which consists of $n$ copies of $K_{k+1}$ all sharing a common $K_k$. In particular, for every fixed integer $k\ge 2$ and for any real $c>1$, let $N=c2^k n$. Then for any real $\gamma>0$, \[ \lim_{n\to \infty} \Pr(G(N,p)\to B_n^{(k)})= \left\{ \begin{array}{cl} 0 & \mbox{if $p\le\frac{1}{c^{1/k}}(1-\gamma)$,} \\ 1 & \mbox{if $p\ge\frac{1}{c^{1/k}}(1+\gamma)$}. \end{array} \right. \] This implies that $r(B_n^{(k)},B_n^{(k)})=2^kn+o(n)$, and hence especially extends the work of Conlon (2019) and the follow-up work of Conlon, Fox and Wigderson (2022) on book Ramsey numbers.