Optimal constant in Shearer’s independent-set bound

Determine the best possible asymptotic constant in Shearer’s lower bound stating that every triangle-free graph on n vertices with average degree d has an independent set of size at least (1+o(1))n\log d/d.

Background

Shearer’s bound gives the best known upper bound on R(3,k). The paper explains that the bound is sharp up to a factor of 2+o(1) when compared with suitable random-graph constructions, and that improving it by any constant factor would have major consequences for the asymptotic determination of R(3,k).

References

It is a notoriously difficult open problem to improve Shearer's bound by any constant factor and it remains a complete mystery as to what the best possible constant in eq:shearer is.

— 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)”