Burr–Erdős conjecture for diagonal tree Ramsey numbers

Prove that for every tree T on n vertices, the diagonal Ramsey number R(T,T) is at most 2n−2 when n is even and at most 2n−3 when n is odd.

Background

The paper introduces classical Ramsey theory through the study of diagonal Ramsey numbers for trees. It presents the Burr–Erdős conjecture as a major conjecture in the area, noting that only a slightly weaker asymptotic form had been proved by Zhao for sufficiently large n. Although this conjecture is not about zero-sum Ramsey numbers modulo 3, it is explicitly stated as an unresolved conjecture in the paper.

References

Much efforts have also been invested in studying the Ramsey numbers of trees, with one of the main conjectures in this area being the following one, due to Burr and Erdős.

— On zero-sum Ramsey numbers modulo 3  (2502.03864 - Caro et al., 6 Feb 2025) in Introduction, Conjecture (Burr–Erdős Conjecture)