Constructions for Nearly Diagonal Book Ramsey Numbers

The Ramsey number R(G₁,G₂) is the smallest positive integer n such that for every graph H on n vertices, either H contains a copy of G₁ or the complement H contains a copy of G₂. Ramsey numbers involving the book graphs Bₙ, which consist of n triangles sharing a common edge, have attracted recent attention, especially in the ``nearly diagonal'' case R(Bₙ₋₁,Bₙ). Rousseau and Sheehan showed the upper bound R(Bₙ₋₁,Bₙ) ≤ 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(Bₙ₋₁,Bₙ) = 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.