PreprintarXiv licencearXiv
The Ramsey threshold for trees versus odd cycles
Abstract
A longstanding fundamental problem of Burr, Erdős, Faudree, Rousseau and Schelp (\emph{Trans. Amer. Math. Soc.}, 1982) is to determine the exact value of the least integer f(m), for odd m≥3, such that every tree Tn on n≥ f(m) vertices satisfies R(Tn,Cm)=2n−1. We settle this problem for all sufficiently large odd m. Indeed, we establish f(m)=⌈ (2m−1)/3 ⌉ for all such m, where the lower bound follows from a result by Faudree, Lawrence, Parsons and Schelp. This also confirms a conjecture of Huang, Zhang and Chen for all such m.
Related
- Off-diagonal Ramsey numbers for 3-uniform tight hypergraph cycles
- The three-colour bipartite Ramsey number of sufficiently long even cycles
- Ramsey--Turán Factors of Non-directed Oriented Cycles in Oriented Graphs
- Ordered Trees with Linear Off-Diagonal Ramsey Numbers
- Anti-Ramsey Number of Intersecting Odd Cycles
BibTeX
@misc{lin2026ramsey,
title = {{The Ramsey threshold for trees versus odd cycles}},
author = {Qizhong Lin and Chunlin You},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.00944},
eprint = {2609.00944},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.00944},
}
RIS
TY - UNPB
TI - The Ramsey threshold for trees versus odd cycles
AU - Qizhong Lin
AU - Chunlin You
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2609.00944
UR - https://arxiv.org/abs/2609.00944
AB - A longstanding fundamental problem of Burr, Erdős, Faudree, Rousseau and Schelp (\emph{Trans. Amer. Math. Soc.}, 1982) is to determine the exact value of the least integer f(m), for odd m≥3, such that every tree T_n on n≥ f(m) vertices satisfies R(T_n,C_m)=2n−1. We settle this problem for all sufficiently large odd m. Indeed, we establish f(m)=⌈ (2m−1)/3 ⌉ for all such m, where the lower bound follows from a result by Faudree, Lawrence, Parsons and Schelp. This also confirms a conjecture of Huang, Zhang and Chen for all such m.
ER -