A proof of a Frankl-Kupavskii conjecture on intersecting families

A family $\mathcal{F}\subset \mathcal{P}(n)$ is $r$-wise $k$-intersecting if $\mathrm{\mid }{A}_1\cap \cdots \cap A_r\mathrm{\mid }\ge k$ for any $A_1,\dots ,A_r\in \mathcal{F}$. It is easily seen that if $\mathcal{F}$ is $r$-wise $k$-intersecting for $r\ge 2$, $k\ge 1$ then $\mathrm{\mid }\mathcal{F}\mathrm{\mid }\le 2^{n-1}$. 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 $n$.