The Ramsey number of books

TL;DR

This paper determines the asymptotic Ramsey number for books, showing that in any two-colouring of a complete graph, a monochromatic book structure appears with a number of extensions close to the theoretical maximum, confirming longstanding conjectures.

Contribution

It establishes the asymptotic value of the Ramsey number for books, answering a question of Erdős et al. and confirming a conjecture of Thomason.

Findings

Ramsey number for books is approximately 2^k n

In any two-colouring, monochromatic books extend to many copies

Results are asymptotically optimal, matching random colourings

Abstract

We show that in every two-colouring of the edges of the complete graph $K_N$ there is a monochromatic $K_k$ which can be extended in at least $(1 + o_k(1))2^{-k}N$ ways to a monochromatic $K_{k+1}$. This result is asymptotically best possible, as may be seen by considering a random colouring. Equivalently, defining the book $B_n^{(k)}$ to be the graph consisting of $n$ copies of $K_{k+1}$ all sharing a common $K_k$, we show that the Ramsey number $r(B_n^{(k)}) = 2^k n + o_k(n)$. In this form, our result answers a question of Erd\H{o}s, Faudree, Rousseau and Schelp and establishes an asymptotic version of a conjecture of Thomason.