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arXiv · 1808.03157

The Ramsey number of books

David Conlon
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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 Erdős, Faudree, Rousseau and Schelp and establishes an asymptotic version of a conjecture of Thomason.

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David Conlon. 2019-10-23. The Ramsey number of books. https://doi.org/10.19086/aic.10808

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