A linear upper bound on the zero-sum Ramsey number of forests in $\mathbb{Z}_p$
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866917353844899840 |
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| author | Colucci, Lucas D'Emidio, Marco |
| author_facet | Colucci, Lucas D'Emidio, Marco |
| contents | Let $m$ be a positive integer and let $G$ be a graph. The zero-sum Ramsey number $R(G,\mathbb{Z}_m)$ is the least integer $N$ (if it exists) such that for every edge-coloring $χ\, : \, E(K_N) \, \rightarrow \, \mathbb{Z}_m$ one can find a copy of $G$ in $K_N$ such that $\sum_{e \, \in \, E(G)}{χ(e)} \, = \, 0$.
In this paper, we show that, for every prime $p$, $$R(F,\mathbb{Z}_p)\leq n+9p-12$$ for every forest $F$ in $n\geq 3p^2-12p+11$ vertices with $p\mid e(F)$ without isolated vertices. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_06229 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A linear upper bound on the zero-sum Ramsey number of forests in $\mathbb{Z}_p$ Colucci, Lucas D'Emidio, Marco Combinatorics Let $m$ be a positive integer and let $G$ be a graph. The zero-sum Ramsey number $R(G,\mathbb{Z}_m)$ is the least integer $N$ (if it exists) such that for every edge-coloring $χ\, : \, E(K_N) \, \rightarrow \, \mathbb{Z}_m$ one can find a copy of $G$ in $K_N$ such that $\sum_{e \, \in \, E(G)}{χ(e)} \, = \, 0$. In this paper, we show that, for every prime $p$, $$R(F,\mathbb{Z}_p)\leq n+9p-12$$ for every forest $F$ in $n\geq 3p^2-12p+11$ vertices with $p\mid e(F)$ without isolated vertices. |
| title | A linear upper bound on the zero-sum Ramsey number of forests in $\mathbb{Z}_p$ |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2512.06229 |