Book Ramsey numbers conjecture

Prove or refute the identity \(R(B_{n-1},B_n)=4n-1\) for every positive integer n, equivalently establish the matching lower bound by constructing a red–blue coloring of \(K_{4n-2}\) containing neither a red \(B_{n-1}\) nor a blue \(B_n\).

Background

The book graph B_k consists of k triangles sharing a common edge. The paper states the general identity as a conjecture and notes that the upper bound 4n−1 is known for all n.

The unresolved part is the matching lower bound. The paper’s three infinite families prove the conjecture for additional parameter values, but do not establish it for every positive integer n.