Small Ramsey numbers for books, wheels, and generalizations
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866917274125860864 |
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| author | Lidický, Bernard McKinley, Gwen Pfender, Florian Van Overberghe, Steven |
| author_facet | Lidický, Bernard McKinley, Gwen Pfender, Florian Van Overberghe, Steven |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_07285 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Small Ramsey numbers for books, wheels, and generalizations Lidický, Bernard McKinley, Gwen Pfender, Florian Van Overberghe, Steven Combinatorics 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. |
| title | Small Ramsey numbers for books, wheels, and generalizations |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2407.07285 |