Exact Ramsey threshold for all odd cycle lengths

Determine whether the Ramsey threshold for every odd integer m≥5 satisfies f(m)=⌈(2m−1)/3⌉, where f(m) is the least integer such that R(T_n,C_m)=2n−1 for every n≥f(m) and every n-vertex tree T_n.

Background

The paper studies the least function f(m) for which every n-vertex tree T_n has Ramsey number R(T_n,C_m)=2n−1 against an odd cycle C_m. A path obstruction gives the lower bound f(m)≥⌈(2m−1)/3⌉. Huang, Zhang, and Chen conjectured that this lower bound is always tight for odd m≥5.

The main theorem proves the conjectured equality only for all sufficiently large odd m. Consequently, the conjecture remains unresolved for the finitely many smaller odd values not covered by the theorem.

References

Huang, Zhang and Chen to conjecture that the path obstruction is the only obstruction, i.e., that equality holds for every odd $m\ge5$. For every odd integer $m\ge5$, $f(m)=\left\lceil\frac{2m-1}{3}\right\rceil$.

— The Ramsey threshold for trees versus odd cycles  (2609.00944 - Lin et al., 1 Sep 2026) in Conjecture 1 (labeled Conjecture~\ref{conj:exact-threshold}), Section 1, Introduction