Papers › Small Ramsey numbers for books, wheels, and generalizations
Small Ramsey numbers for books, wheels, and generalizations
Bernard Lidický, Gwen McKinley, Florian Pfender, Steven Van Overberghe
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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(W₅, W₇) = 15, matching upper and lower bounds giving R(W₅, W₉) = 18, R(B₂, B₈) = 21, and R(B₃, B₇) = 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,Kₛ,t) denote the minimum number of vertices n such that any r-edge-coloring of Kₙ has a copy of Kₛ with at most t colors. We establish GR(3,K₄,2) = 10, GR(4,K₄,3) = 10, and some additional bounds.
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