A proof of a Frankl-Kupavskii conjecture on intersecting families

Abstract

A family $\mathcal{F} \subset \mathcal{P}(n)$ is $r$-wise $k$-intersecting if $|A_1 \cap \dots \cap A_r| \geq k$ for any $A_1, \dots, A_r \in \mathcal{F}$. It is easily seen that if $\mathcal{F}$ is $r$-wise $k$-intersecting for $r \geq 2$, $k \geq 1$ then $|\mathcal{F}| \leq 2^{n-1}$. The problem of determining the maximal size of a family $\mathcal{F}$ that is both $r_1$-wise $k_1$-intersecting and $r_2$-wise $k_2$-intersecting was raised in 2019 by Frankl and Kupavskii [1]. They proved the surprising result that, for $(r_1,k_1) = (3,1)$ and $(r_2,k_2) = (2,32)$ then this maximum is at most $2^{n-2}$, and conjectured the same holds if $k_2$ is replaced by $3$. In this paper we shall not only prove this conjecture but we shall also determine the exact maximum for $(r_1,k_1) = (3,1)$ and $(r_2,k_2) = (2,3)$ for all $n$.