The asymptotics of off-diagonal book Ramsey numbers

Abstract

Let $B_n$ be the book graph consisting of $n$ triangles sharing a common edge. We prove upper bounds for $r(B_{\lceil \alpha n\rceil },B_n)$ that asymptotically match the known lower bounds in the remaining range $1/6<\alpha <1/4$. Combined with the known results for the other ranges of $\alpha$, this completely determines the asymptotics of the Ramsey number $r(B_{\lceil \alpha n\rceil },B_n)$ for every fixed $0<\alpha \le 1$: \[ r(B_{\lceilαn\rceil},B_n)=(c(α)+o(1))n, \qquad c(α)= \begin{cases} 2, & 0<α\le1/6,\\[1mm] \displaystyle\frac{3}{1+2p_α^2}, & 1/6<α\leα_0,\\[3mm] (1+\sqrtα)^2, & α_0<α\le1, \end{cases} \] where \(α_0:=\frac{52-16\sqrt3}{121}, \;\; p_α:=\frac{1-\sqrt{α(3-2α)}}{1-2α} \;\; (1/6<α\leα_0).\) Consequently, the random lower bound is asymptotically tight if and only if $\alpha \ge {\alpha }_0$, thereby determining exactly the range of $\alpha$ for which the conjecture of Conlon, Fox and Wigderson holds. The proof develops a defect-compensation framework around a $K_4$-free auxiliary graph.

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