Asymptotic constant for R(3,k)

Prove that the Ramsey number R(3,k) satisfies R(3,k)=(1/2+o(1))k^2/\log k as k tends to infinity.

Background

The paper improves the known lower bound for R(3,k) from (1/4+o(1))k^2/\log k to (1/3+o(1))k^2/\log k, while the best known upper bound remains (1+o(1))k^2/\log k. The authors therefore conjecture that the true asymptotic constant is 1/2.

A forthcoming companion construction based on a Cayley sum graph generated by the sum-free process on \mathbb{F}_2^d is described as potential evidence for the conjecture. The paper also notes that an improved seed step in its own construction might achieve the same constant, but does not establish this.