A new intersection condition in extremal set theory

Abstract

We call a family $\mathcal{F}$ $(3,2,\ell )$-intersecting if $|A\cap B|+|B\cap C|+|C\cap A|\ge \ell$ for all $A$, $B$, $C\in \mathcal{F}$. We try to look for the maximum size of such a family $\mathcal{F}$ in case when $\mathcal{F}\subset \left(\genfrac{}{}{0ex}{}{[n]}{k}\right)$ or $\mathcal{F}\subset 2^{[n]}$. In the uniform case we show that if $\mathcal{F}$ is $(3,2,2)$-intersecting, then $|\mathcal{F}|\le \left(\genfrac{}{}{0ex}{}{n+1}{k-1}\right)+\left(\genfrac{}{}{0ex}{}{n}{k-2}\right)$ and if $\mathcal{F}$ is $(3,2,3)$-intersecting, then $|\mathcal{F}|\le \left(\genfrac{}{}{0ex}{}{n}{k-1}\right)+2\left(\genfrac{}{}{0ex}{}{n}{k-3}\right)+3\left(\genfrac{}{}{0ex}{}{n-1}{k-3}\right)$. For the lower bound we construct a $(3,2,\ell )$-intersecting family and we show that this bound is sharp when $\ell =2$ or $3$ and $n$ is sufficiently large compared to $k$. In the non-uniform case we give an upper bound for a $(3,2,n-x)$-intersecting family, when $n$ is sufficiently large compared to $x$.

15 pages