讲座内容：

For graphs G and H, the Ramsey number r(G,H) is the smallest positive integer N such that any red/blue edge coloring of the complete graph K_N contains either a red G or a blue H. A book B_n is a graph consisting of n triangles all sharing a common edge. Recently, Conlon, Fox and Wigderson (2023) conjecture that for any 0<\alpha<1, the random lower bound r(B_{\lceil\alpha n\rceil},B_n)\ge (\sqrt{\alpha}+1)^2n+o(n) would not be tight. In other words, there exists some constant \beta=\beta(\alpha)>0 such that r(B_{\lceil\alpha n\rceil},B_n)\ge (\sqrt{\alpha}+1)^2n+\beta n for all sufficiently large n. This conjecture clearly holds for every \alpha< 1/6 from an early result of Nikiforov and Rousseau (2005), i.e., for every \alpha< 1/6 and large n, r(B_{\lceil\alpha n\rceil},B_n)=2n+3.

We disprove the conjecture of Conlon et al. (2023). Indeed, we show that the random lower bound is asymptotically tight for every 1/4\leq \alpha\leq 1. Moreover, we show that for any 1/6\leq \alpha\le 1/4 and large n, r(B_{\lceil\alpha n\rceil},B_n)\le\left(\frac 32+3\alpha\right) n+o(n), where the inequality is asymptotically tight when \alpha=1/6 or 1/4. We also give a lower bound of r(B_{\lceil\alpha n\rceil},B_n) for 1/6\le\alpha<\frac{52-16\sqrt{3}}{121}\approx0.2007, showing that the random lower bound is not tight, i.e., the conjecture of Conlon et al. (2023) holds in this interval.

Joint work with Chunchao Fan and Yuanhui Yan.