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

Abstract

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 $1\le i\le r$, $n\notin F_i$ for each $i>r$ and $|F_i|\le |F_j|$ 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.