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 $ρ$. What is the minimum value $a(ρ)$ for which there always exist two non-adjacent vertices such that the density of their common neighborhood is $\leq a(ρ)$? 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 $C_5$, 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 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 $a(ρ)\geqε$ but this finiteness result is somewhat purely existential (the bound is double exponential in $1/ε$).

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