Improved seed construction for the modified triangle-free process

Construct a better seed graph for the modified triangle-free process that achieves a lower-bound constant of 1/2+o(1) for R(3,k).

Background

The paper’s construction combines a blown-up random seed graph with a modified triangle-free process and obtains the constant 1/3. The authors observe that optimizing the process suggests a constant 1/2 may be possible if the seed step is improved, but they do not provide such a seed.

References

It also seems possible that one could provide a better ``seed step'' in our process from this paper to obtain a constant of $1/2+o(1)$, although we were unable to find such a seed.

— A new lower bound for the Ramsey numbers $R(3,k)$  (2505.13371 - Campos et al., 19 May 2025) in Section 1, subsection “The asymptotic value of R(3,k)”