Small Ramsey numbers for books, wheels, and generalizations

Abstract

In this work, we give several new upper and lower bounds on Ramsey numbers for books and wheels, including a tight upper bound establishing $R(W_5, W_7) = 15$, matching upper and lower bounds giving $R(W_5, W_9) = 18$, $R(B_2, B_8) = 21$, and $R(B_3, B_7) = 20$, and a number of additional tight lower bounds for books. We use a range of different methods: flag algebras, local search, bottom-up generation, and enumeration of polycirculant graphs. We also explore generalized Ramsey numbers using similar methods. Let $GR(r,K_s,t)$ denote the minimum number of vertices $n$ such that any $r$-edge-coloring of $K_n$ has a copy of $K_s$ with at most $t$ colors. We establish $GR(3,K_4,2) = 10, GR(4,K_4,3) = 10$, and some additional bounds.