An extremal problem motivated by triangle-free strongly regular graphs

We introduce the following combinatorial problem. Let G be a triangle-free regular graph with edge density ρ. (In this paper all densities are normalized by n,n22 etc. rather than by n−1,(n2),…) What is the minimum value a(ρ) for which there always exist two non-adjacent vertices such that the density of their common neighbourhood is ≤a(ρ)? We prove a variety of upper bounds on the function a(ρ) that are tight for the values ρ=2/5,5/16,3/10,11/50, with C5, Clebsch, Petersen and Higman-Sims being respective extremal configurations. Our proofs are entirely combinatorial and are largely based on counting densities in the style of flag algebras. For small values of ρ, our bound attaches a combinatorial meaning to so-called Krein conditions that might be interesting in its own right. We also prove that for any ϵ>0 there are only finitely many values of ρ with a(ρ)≥ϵ but this finiteness result is somewhat purely existential (the bound is double exponential in 1/ϵ).