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arXiv · 2407.07285

Small Ramsey numbers for books, wheels, and generalizations

Bernard Lidický·Gwen McKinley·Florian Pfender·Steven Van Overberghe
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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.

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Bernard Lidický, Gwen McKinley, Florian Pfender, Steven Van Overberghe. 2024-07-10. Small Ramsey numbers for books, wheels, and generalizations. https://doi.org/10.37236/13577

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