Multiply-intersecting families revisited

Abstract

Motivated by the Frankl's results in [P. Frankl, Multiply-intersecting families, J. Combin. Theory B 53 (1991) 195–234], we consider some problems concerning the maximum size of multiply-intersecting families with additional conditions. Among other results, we show the following version of the Erdős–Ko–Rado theorem: for all $r⩾5$ and $1⩽t⩽2^{r+1}-3r-1$ there exist positive constants ε and $n_0$ such that if $n>n_0$ and $|\frac{k}{n}-\frac{1}{2}|<\epsilon$ then r-wise t-intersecting k-uniform families on n vertices have size at most $\max \{\left(\begin{array}{c}\hfill n-t\hfill \\ \hfill k-t\hfill \end{array}\right),(t+r)\left(\begin{array}{c}\hfill n-t-r\hfill \\ \hfill k-t-r+1\hfill \end{array}\right)+\left(\begin{array}{c}\hfill n-t-r\hfill \\ \hfill k-t-r\hfill \end{array}\right)\}$.