PreprintCC BY 4.0arXiv
On a problem concerning ℤ3-Ramsey numbers
Abstract
For a graph G with 3| e(G), let R(G,ℤ3) denote the least integer N such that every edge weighting w:E(KN)→ℤ3 admits a copy of G whose edge weights sum to zero. We prove that every n-vertex graph G with n≥6 and 3| e(G) satisfies R(G,ℤ3)≤ n+5. Moreover, the stronger bound R(G,ℤ3)≤ n+4 holds whenever G contains an induced P4 or an induced 2K2. Consequently, we resolve a problem posed by Caro and Mifsud: every graph G with 3| e(G) satisfies R(G,ℤ3)≤ |V(G)|+8, with equality if and only if G≅ K3.
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BibTeX
@misc{chi2026problem,
title = {{On a problem concerning $\mathbb Z_3$-Ramsey numbers}},
author = {Cheng Chi and Jialin He},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27788},
eprint = {2609.27788},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27788},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On a problem concerning ℤ_3-Ramsey numbers
AU - Cheng Chi
AU - Jialin He
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.27788
UR - https://arxiv.org/abs/2609.27788
AB - For a graph G with 3| e(G), let R(G,ℤ_3) denote the least integer N such that every edge weighting w:E(K_N)→ℤ_3 admits a copy of G whose edge weights sum to zero. We prove that every n-vertex graph G with n≥6 and 3| e(G) satisfies R(G,ℤ_3)≤ n+5. Moreover, the stronger bound R(G,ℤ_3)≤ n+4 holds whenever G contains an induced P_4 or an induced 2K_2. Consequently, we resolve a problem posed by Caro and Mifsud: every graph G with 3| e(G) satisfies R(G,ℤ_3)≤ |V(G)|+8, with equality if and only if G≅ K_3.
ER -