Sharp Ramsey thresholds for large books

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 $N=c{2}^{k}n$. Then for any real ${\hat{I}}^3>0$, \[ \lim_{n\to \infty} \Pr(G(N,p)\to B_n^{(k)})= \left\{ \begin{array}{cl} 0 & \mbox{if $p\le \frac{1}{c^{1/k}}(1-{\hat{I}}^3)$,} \\ 1 & \mbox{if $p\ge \frac{1}{c^{1/k}}(1+{\hat{I}}^3)$}. \end{array} \right. \] This implies that $r(B_n^{(k)},B_n^{(k)})=2^{k}n+o(n)$, and hence especially extends the work of Conlon (2019) and the follow-up work of Conlon, Fox and Wigderson (2022) on book Ramsey numbers.

15 pages