A new intersection condition in extremal set theory

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 $x$.