Upper bounds for the size of ordered $L$-intersecting set systems

A family \mbox{\cal F}=\{F_1,\ldots,F_m\} of subsets of $[n]$ is said to be ordered, if there exists an $1\le r\le m$ index such that $n\in F_i$ for each $1\le i\le r$, $n\mathrm{\notin }{F}_i$ for each $i>r$ and $\mathrm{\mid }{F}_i\mathrm{\mid }\le \mathrm{\mid }{F}_j\mathrm{\mid }$ for each $1\le i<j\le m$. Our main result is a new upper bound for the size of ordered $L$-intersecting set systems.