On a conjecture of Conlon, Fox, and Wigderson

For graphs G and H , the Ramsey number $r(G,H)$ is the smallest positive integer N such that any red/blue edge colouring of the complete graph KN contains either a red G or a blue H . A book Bₙ is a graph consisting of n triangles all sharing a common edge. Recently, Conlon, Fox, and Wigderson conjectured that for any 0< α < 1 , the random lower bound r(Bα n,Bₙ)≥ (√α +1)²n+o(n) is not tight. In other words, there exists some constant β > (√α +1)² such that r(Bα n,Bₙ)≥ β n for all sufficiently large n . This conjecture holds for every α < 1/6 by a result of Nikiforov and Rousseau from 2005, which says that in this range r(Bα n,Bₙ)=2n+3 for all sufficiently large n . We disprove the conjecture of Conlon, Fox, and Wigderson. Indeed, we show that the random lower bound is asymptotically tight for every 1/4≤ α ≤ 1 . Moreover, we show that for any 1/6≤ α ≤ 1/4 and large n , r(Bα n, Bₙ)≤ ( 32+3α ) n+o(n) , where the inequality is asymptotically tight when α =1/6 or $1/4$ . We also give a lower bound of r(Bα n, Bₙ) for 1/6≤ α < 52-16√3121≈ 0.2007 , showing that the random lower bound is not tight, i.e., the conjecture of Conlon, Fox, and Wigderson holds in this interval.