Optimal constant in Shearer’s theorem

Determine the correct asymptotic constant in the independence-number lower bound for triangle-free graphs of large average degree.

Background

Shearer’s theorem gives an independent set of size approximately n log d/d in a triangle-free graph with average degree d. The paper states that the exact multiplicative constant is unknown and that it is not clear whether either the currently known upper or lower bound is sharp.

References

We pause to remark that it is a major open problem to determine the correct constant in Theorem~\ref{thm:sheaer}. It is unclear which if either the upper bound or lower bound is sharp.

— Probabilistic combinatorics at exponentially small scales  (2512.15077 - Sahasrabudhe, 17 Dec 2025) in Section 3, subsection “Towards an optimal version of Shearer’s theorem”