A new intersection condition in extremal set theory

We call a family $\mathcal{F}$ $(3,2,\mathrm{\ell })$-intersecting if $\mathrm{\mid }A\cap B\mathrm{\mid }+\mathrm{\mid }B\cap C\mathrm{\mid }+\mathrm{\mid }C\cap A\mathrm{\mid }\ge \mathrm{\ell }$ 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 $\mathcal{F}\subset \left(\genfrac{}{}{0px}{}{[n]}{k}\right)$ or $\mathcal{F}\subset 2^{[n]}$. In the uniform case we show that if $\mathcal{F}$ is $(3,2,2)$-intersecting, then $\mathrm{\mid }\mathcal{F}\mathrm{\mid }\le \left(\genfrac{}{}{0px}{}{n+1}{k-1}\right)+\left(\genfrac{}{}{0px}{}{n}{k-2}\right)$ and if $\mathcal{F}$ is $(3,2,3)$-intersecting, then $\mathrm{\mid }\mathcal{F}\mathrm{\mid }\le \left(\genfrac{}{}{0px}{}{n}{k-1}\right)+2\left(\genfrac{}{}{0px}{}{n}{k-3}\right)+3\left(\genfrac{}{}{0px}{}{n-1}{k-3}\right)$. For the lower bound we construct a $(3,2,\mathrm{\ell })$-intersecting family and we show that this bound is sharp when $\mathrm{\ell }=2$ 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 $x$.