Triangle Ramsey numbers of complete graphs
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866918157679067136 |
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| author | Fox, Jacob Tidor, Jonathan Zhang, Shengtong |
| author_facet | Fox, Jacob Tidor, Jonathan Zhang, Shengtong |
| contents | A graph is $H$-Ramsey if every two-coloring of its edges contains a monochromatic copy of $H$. Define the $F$-Ramsey number of $H$, denoted by $r_F(H)$, to be the minimum number of copies of $F$ in a graph which is $H$-Ramsey. This generalizes the Ramsey number and size Ramsey number of a graph. Addressing a question of Spiro, we prove that \[r_{K_3}(K_t)=\binom{r(K_t)}{3}\] for all sufficiently large $t$. We do so through a result on graph coloring: there exists an absolute constant $K$ such that every $r$-chromatic graph where every edge is contained in at least $K$ triangles must contain at least $\binom{r}{3}$ triangles in total. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2312_06895 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Triangle Ramsey numbers of complete graphs Fox, Jacob Tidor, Jonathan Zhang, Shengtong Combinatorics A graph is $H$-Ramsey if every two-coloring of its edges contains a monochromatic copy of $H$. Define the $F$-Ramsey number of $H$, denoted by $r_F(H)$, to be the minimum number of copies of $F$ in a graph which is $H$-Ramsey. This generalizes the Ramsey number and size Ramsey number of a graph. Addressing a question of Spiro, we prove that \[r_{K_3}(K_t)=\binom{r(K_t)}{3}\] for all sufficiently large $t$. We do so through a result on graph coloring: there exists an absolute constant $K$ such that every $r$-chromatic graph where every edge is contained in at least $K$ triangles must contain at least $\binom{r}{3}$ triangles in total. |
| title | Triangle Ramsey numbers of complete graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2312.06895 |