Lower Bounds for Book Ramsey Numbers

TL;DR

This paper establishes new bounds for Ramsey numbers involving book graphs, utilizing combinatorial constructions, computational methods, and symmetry enumeration to advance understanding of these graph parameters.

Contribution

It introduces new bounds for $R(B_{n-1},B_n)$ using block-circulant graphs and improves bounds for other pairs via SAT and IP solvers, also enumerating critical graphs.

Findings

Proved $R(B_{n-1},B_n) = 4n-1$ for an infinite family of $n$

Improved bounds for several $R(B_r,B_s)$ values

Enumerated critical graphs for small $r,s$ using SAT modulo symmetries

Abstract

We prove new bounds for Ramsey numbers for book graphs $B_n$. In particular, we show that $R(B_{n-1},B_n) = 4n-1$ for an infinite family of $n$ using a block-circulant construction similar to Paley graphs. We obtain improved bounds for several other values of $R(B_r,B_s)$ using different block-circulant graphs from SAT and integer programming (IP) solvers. Finally, we enumerate the number of critical graphs for $R(B_r,B_s)$ for small $r$ and $s$ using SAT modulo symmetries (SMS).