The Exact Erdős-Ko-Rado Theorem for 3-wise $t$-intersecting uniform families

Abstract

Let $\mathcal{F}$ be a family of $k$-element subsets of $\{1,2,\dots ,n\}$. For $t\ge 1$, we say that $\mathcal{F}$ is {\it 3-wise $t$-intersecting} if $|F_1\cap F_2\cap F_3|\ge t$ for all $F_1,F_2,F_3\in \mathcal{F}$. In the present paper, we prove that if $\mathcal{F}$ is 3-wise $t$-intersecting and $n\ge \frac{\sqrt{4t+9}-1}{2}k$, $k>t\ge 46$, then $|\mathcal{F}|\le \left(\genfrac{}{}{0ex}{}{n-t}{k-t}\right)$. The restriction on $n$ is asymptotically best possible. The corresponding result for non-trivial 3-wise $t$-intersecting families is obtained as well for $n\ge \frac{\sqrt{4t+9}-1}{2}k$ and $k>t\ge 55$.