An extension of the Erdős–Ko–Rado Theorem

Abstract

A family $\mathcal{G}\subset 2^{[n]}$ is called r-wise t-intersecting if $|G_1\cap \cdots \cap G_r|⩾t$ holds for all $G_1,\dots ,G_r\in \mathcal{G}$. We report some results concerning the maximum size of r-wise t-intersecting families with additional conditions. For example, any k-uniform r-wise t-intersecting family on 2k vertices has size at most $\left(\begin{array}{c}\hfill 2k-1\hfill \\ \hfill k-1\hfill \end{array}\right)$ for $r⩾7$, $1⩽t⩽2^r-r-1$ and $k>k_0(r)$.