The Exact Erd\H{o}s-Ko-Rado Theorem for 3-wise $t$-intersecting uniform families

TL;DR

This paper establishes the exact maximum size of 3-wise t-intersecting uniform families of sets for large enough n, extending the Erdős-Ko-Rado theorem to this specific intersection condition.

Contribution

It provides the first exact bounds for the size of 3-wise t-intersecting families, including non-trivial cases, for sufficiently large n, with optimal restrictions on n.

Findings

Maximum size of 3-wise t-intersecting families is inom{n-t}{k-t} for large n.

The bounds are asymptotically optimal.

Results include non-trivial families for t  55.

Abstract

Let $\mathcal{F}$ be a family of $k$-element subsets of $\{1,2,\ldots,n\}$. For $t\geq 1$, we say that $\mathcal{F}$ is {\it 3-wise $t$-intersecting} if $|F_1\cap F_2\cap F_3|\geq t$ 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 $n\geq \frac{\sqrt{4t+9}-1}{2}k$, $k>t\geq 46$, then $|\mathcal{F}|\leq \binom{n-t}{k-t}$. The restriction on $n$ is asymptotically best possible. The corresponding result for non-trivial 3-wise $t$-intersecting families is obtained as well for $n\geq \frac{\sqrt{4t+9}-1}{2}k$ and $k>t\geq 55$.