On zero-sum Ramsey numbers modulo 3
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arXiv
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| Format: | Preprint |
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2025
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| author | Caro, Yair Mifsud, Xandru |
| author_facet | Caro, Yair Mifsud, Xandru |
| contents | We start with a systematic study of the zero-sum Ramsey numbers. For a graph $G$ with $0 \ (\!\!\!\!\mod 3)$ edges, the zero-sum Ramsey number is defined as the smallest positive integer $R(G, \mathbb{Z}_3)$ such that for every $n \geq R(G, \mathbb{Z}_3)$ and every edge-colouring $f$ of $K_n$ using $\mathbb{Z}_3$, there is a zero-sum copy of $G$ in $K_n$ coloured by $f$, that is: $\sum_{e \in E(G)} f(e) \equiv 0 \ (\!\!\!\!\mod 3)$.
Only sporadic results are known for these Ramsey numbers, and we discover many new ones. In particular we prove that for every forest $F$ on $n$ vertices and with $0 \ (\!\!\!\!\mod 3)$ edges, $R(F, \mathbb{Z}_3) \leq n+2$, and this bound is tight if all the vertices of $F$ have degrees $1 \ (\!\!\!\!\mod 3)$. We also determine exact values of $R(T, \mathbb{Z}_3)$ for infinite families of trees. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_03864 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On zero-sum Ramsey numbers modulo 3 Caro, Yair Mifsud, Xandru Combinatorics 05C15, 05C35, 05C55, 05D10 We start with a systematic study of the zero-sum Ramsey numbers. For a graph $G$ with $0 \ (\!\!\!\!\mod 3)$ edges, the zero-sum Ramsey number is defined as the smallest positive integer $R(G, \mathbb{Z}_3)$ such that for every $n \geq R(G, \mathbb{Z}_3)$ and every edge-colouring $f$ of $K_n$ using $\mathbb{Z}_3$, there is a zero-sum copy of $G$ in $K_n$ coloured by $f$, that is: $\sum_{e \in E(G)} f(e) \equiv 0 \ (\!\!\!\!\mod 3)$. Only sporadic results are known for these Ramsey numbers, and we discover many new ones. In particular we prove that for every forest $F$ on $n$ vertices and with $0 \ (\!\!\!\!\mod 3)$ edges, $R(F, \mathbb{Z}_3) \leq n+2$, and this bound is tight if all the vertices of $F$ have degrees $1 \ (\!\!\!\!\mod 3)$. We also determine exact values of $R(T, \mathbb{Z}_3)$ for infinite families of trees. |
| title | On zero-sum Ramsey numbers modulo 3 |
| topic | Combinatorics 05C15, 05C35, 05C55, 05D10 |
| url | https://arxiv.org/abs/2502.03864 |