An Erd\H{o}s--Ko--Rado theorem for cross-intersecting families in the Euclidean inner product

Abstract

Let $\binom{[n]}{k}$ be the set of all $k$-element subsets of the set $\{1,\ldots,n\}$ and let $\mathcal A,\mathcal B \subseteq \binom{[n]}{k}$ be two cross-intersecting families, that is, $A\cap B\neq \emptyset$ for any $A\in \mathcal A$ and $B\in \mathcal B$. The classical cross-intersecting version of the Erd\H{o}s--Ko--Rado theorem, due to Pyber and Matsumoto--Tokushige, states that if $n\geq 2k$, then $|\mathcal A||\mathcal B|\leq \binom{n-1}{k-1}^2,$ where the equality holds for $n>2k$ if and only if $\mathcal A=\mathcal B$ is a star. In the present paper, we first give a stability result of this theorem by using Filmus's FKN theorem on the slice and linear algebra method as follows: There exists a constant $C>1$ such that if $n\geq 2.07k$ and $|\mathcal A||\mathcal B|\geq (1-\epsilon)\binom{n-1}{k-1}^2$, where $\epsilon\leq \frac{k^2}{C^2 n ^2 }$, then there is a star $\mathcal{S}$ such that $|\mathcal{S} \Delta \mathcal A|\leq C \epsilon \binom{n}{k}$ and $|\mathcal{S} \Delta \mathcal B|\leq C \epsilon \binom{n}{k}.$ Moreover, based on this stability result and the eigenvalues of the matrices of the Johnson scheme, we present an Erd\H{o}s--Ko--Rado theorem for cross-intersecting families in the Euclidean inner product showing that if $n\geq 2k$ and $k\geq d \geq 0$, then $$\big\langle\mathbf{v}_d(\mathcal A),\mathbf{v}_d(\mathcal B)\big\rangle \leq \frac{\binom{k}{d}\binom{k-1}{d}}{\binom{n-1}{d}}\binom{n-1}{k-1}^2 +\binom{k-1}{d-1} \binom{n-d-1}{k-d}\binom{n-1}{k-1},$$ together with uniqueness and a corresponding stability result, where $\mathbf{v}_d(\mathcal A) \in \mathbb R^{\binom{[n]}{d}}$ is the $d$-degree vector of $\mathcal A$ whose $U$-entry is the number of members in $\mathcal A$ containing $U$.