Constructions for Nearly Diagonal Book Ramsey Numbers

Abstract

The Ramsey number $R(G_1,G_2)$ is the smallest positive integer $n$ such that for every graph $H$ on $n$ vertices, either $H$ contains a copy of $G_1$ or the complement $\overline H$ contains a copy of $G_2$. Ramsey numbers involving the book graphs $B_n$, which consist of $n$ triangles sharing a common edge, have attracted recent attention, especially in the ``nearly diagonal'' case $R(B_{n-1},B_n)$. Rousseau and Sheehan showed the upper bound $R(B_{n-1},B_n) \le 4n-1$ nearly fifty years ago, but no general matching lower bound construction is known. This paper gives several new infinite families of $n$ for which $R(B_{n-1},B_n) = 4n-1$. Some of the main results in this paper were found with assistance from the Station, a multi-agent AI system. These results also appeared in a larger work on autonomous mathematical discovery by the authors and Du in arxiv:2608.23691. This paper is a follow-up to that work: we give complete proofs of the book Ramsey results and discuss their connections to objects in combinatorial design theory.