We call a family $\mathcal{ F}$ $(3,2,\ell)$-intersecting if $|A \cap B|+|B \cap C|+|C \cap A| \geq \ell$ 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 $\mathcal{ F} \subset { [n] \choose k}$ or $\mathcal{ F} \subset 2^{ [n]}$. In the uniform case we show that if $\mathcal{ F}$ is $(3,2,2)$-intersecting, then $\vert \mathcal{ F} \vert \leq { n+1 \choose k-1}+{ n \choose k-2}$ and if $\mathcal{ F}$ is $(3,2,3)$-intersecting, then $|\mathcal{ F}| \leq { n \choose k-1} + 2 { n \choose k-3} + 3 { n-1 \choose k-3}$. 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 $x$.
Categories: math.CO