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 $n,\frac{n^2}{2}$ etc. rather than by $n-1,\left(\begin{array}{c}n\\ 2\end{array}\right),\dots$) What is the minimum value $a(\rho )$ for which there always exist two non-adjacent vertices such that the density of their common neighbourhood is $\le a(\rho )$? We prove a variety of upper bounds on the function $a(\rho )$ that are tight for the values $\rho =2/5,5/16,3/10,11/50$, 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 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 $a(\rho )\ge ϵ$ but this finiteness result is somewhat purely existential (the bound is double exponential in $1/ϵ$).

我们引入以下组合问题。令G为边密度为ρ的无三角形正则图。（在本文中，所有密度都归一化为$n,\frac{n^2}{2}$等等而不是通过$n-1,\left(\begin{array}{c}n\\ 2\end{array}\right),\dots$) 最小值是多少$\mathrm{一种}(\rho )$总是存在两个不相邻的顶点，使得它们的公共邻域的密度为$\le \mathrm{一种}(\rho )$? 我们证明了函数的多种上界$\mathrm{一种}(\rho )$对值很紧$\rho =2/5,5/16,3/10,11/50$， 和$C_5$, Clebsch, Petersen 和 Higman-Sims 是各自的极值配置。我们的证明完全是组合的，并且主要基于标志代数风格的计数密度。对于较小的ρ值，我们的界限将组合含义附加到所谓的 Kerin 条件，这可能本身就很有趣。我们还证明对于任何$\epsilon >0$只有有限多个ρ值$\mathrm{一种}(\rho )\ge \epsilon$但是这种有限性结果在某种程度上是纯粹存在的（界限是双指数的$1/\epsilon$）。