New bounds for some small multicolor Ramsey numbers
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866908518909476864 |
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| author | Wesley, William J. |
| author_facet | Wesley, William J. |
| contents | The Ramsey number $R(G_1,\dots,G_k)$ is the smallest $n$ such that every $k$-coloring of the edges of $K_n$ contains a monochromatic copy of $G_i$ in color $i$. Ramsey numbers are challenging to compute, and few are known exactly. We use Boolean satisfiability (SAT) solvers to search for structured colorings that give lower bounds, and we show $R(K_4,K_4-e,K_4-e) \ge 35$ and $R(K_3,K_4,C_4,C_4) \ge 49$. Moreover, we tighten some recent upper bounds for multicolor Ramsey numbers for cycles and show $R(C_3,C_6,C_6) = R(C_5,C_6,C_6) = 15$. Finally, we enumerate critical graphs for the numbers $R(C_4,K_{1,s})$ and $R(C_6,K_{1,s})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_03784 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | New bounds for some small multicolor Ramsey numbers Wesley, William J. Combinatorics The Ramsey number $R(G_1,\dots,G_k)$ is the smallest $n$ such that every $k$-coloring of the edges of $K_n$ contains a monochromatic copy of $G_i$ in color $i$. Ramsey numbers are challenging to compute, and few are known exactly. We use Boolean satisfiability (SAT) solvers to search for structured colorings that give lower bounds, and we show $R(K_4,K_4-e,K_4-e) \ge 35$ and $R(K_3,K_4,C_4,C_4) \ge 49$. Moreover, we tighten some recent upper bounds for multicolor Ramsey numbers for cycles and show $R(C_3,C_6,C_6) = R(C_5,C_6,C_6) = 15$. Finally, we enumerate critical graphs for the numbers $R(C_4,K_{1,s})$ and $R(C_6,K_{1,s})$. |
| title | New bounds for some small multicolor Ramsey numbers |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2509.03784 |