Ramsey threshold for trees versus odd cycles

Burr, Erdős, Faudree, Rousseau, and Schelp initiated the problem of finding the least threshold $f(m)$ such that $R(T_n,C_m)=2n-1$ for all sufficiently large $n$, with $m$ odd. The central conjecture predicts $f(m)=\left\lceil(2m-1)/3\right\rceil$ for odd $m\geq 5$.

Known results

• Burr, Erdős, Faudree, Rousseau, and Schelp (2016): $f(m)\leq 756m^{10}$.
• Brennan (2016): $f(m)\leq 25m$.
• Fan and Lin (2025): $f(m)\leq 4m-8$.
• Huang, Zhang, and Chen (2026): $f(m)\leq 2m-4$, with $\left\lceil(2m-1)/3\right\rceil\leq f(m)$; they stated the exact-threshold conjecture.

September 1, 2026 claimed resolution

A preprint identified on September 1, 2026 claims to settle the exact threshold for every sufficiently large odd $m$ and to confirm the Huang--Zhang--Chen conjecture in that range. This claim is unrefereed and has no independent verification in the retrieved material.

Current status (as of September 2026): The threshold is claimed to equal $\left\lceil(2m-1)/3\right\rceil$ for sufficiently large odd $m$, but that claim is unverified and cases outside that range remain open.