Dense triangle-free subgraphs in Ramsey graphs

Determine whether there exists a constant α > 1/2 such that, for every integer r ≥ 2, one can find a graph G satisfying G → (K₃)ᵣ in which every subgraph H ⊆ G contains a triangle-free subgraph H′ ⊆ H with |E(H′)| ≥ α|E(H)|.

Background

Theorem 1.6 establishes that for every bipartite graph F satisfying the stated inseparability condition and every number of colours r ≥ 2, there exists an F-Ramsey graph G such that every subgraph H of G contains an F-free subgraph H′ retaining at least a proportion (ℓ(F) − 1)/(2ℓ(F)) of the edges of H. The authors note that this proportion is less than one half and that the resulting assertion is therefore trivial for graphs with chromatic number at least three.

Question 1.7 asks whether the bound can be improved beyond one half in the special case F = K₃. Specifically, it asks for a uniform constant α > 1/2 such that every subgraph of suitable multicolour Ramsey graphs for triangles contains a triangle-free subgraph retaining at least an α proportion of its edges. The question remains unresolved in the paper.