Exact book Ramsey numbers for adjacent parameter families

For books B(k), consisting of k triangles sharing an edge, when does the Ramsey number R(B(n−1),B(n)) equal 4n−1? Known upper bounds leave the task of constructing matching colorings for more parameters.

Why it matters and what counts as progress

Why it matters: Matching coloring constructions determine when the known book-Ramsey upper bound is exact.

What counts as progress: Give matching colorings for new infinite parameter families or determine exact book Ramsey numbers beyond those families.

How the problem developed

1978 · Bound [1]
C. C. Rousseau and J. Sheehan Established the upper bound R(B_(n−1), B_n) ≤ 4n − 1 and initiated the systematic study of book Ramsey numbers.
2 May 2026 · Construction [2]
David Turturean, using GPT-5.2 Pro and GPT-5.4 Pro for mathematics and Claude Opus 4.6 for coding Reported two additional infinite families and a search program finding constructions for all remaining n ≤ 56 in the benchmark range.
13 Aug 2026 · Construction [3]
Station agents, with an external expert recognizing and generalizing the Yamada–Pott family Reported three infinite families of exact values. The paper says the agents supplied finite constructions and a correlation identity for the Yamada–Pott family, which an outside expert generalized.

AI contribution

Autonomous Mathematical Discovery in an Open-World Multi-Agent Environment

Stephen Chung, Wenyu Du, William J. Wesley

Station v2 Section 6.13 S1: a conference graph with parameters (q,(q-1)/2,(q-5)/4,(q-1)/4) gives R(B_q,B_(q+1))=4q+3. S2: every prime power Q>3 with Q=3 mod 8 gives R(B_((Q-1)/2),B_((Q+1)/2))=2Q+1. Exclude the third Yamada–Pott family requiring human expert synthesis.

AI involvementAI-autonomous

ImportanceNotable

Technical verificationFormal (Lean)

Human understandingFull