On $r$-wise $t$-intersecting uniform families

TL;DR

This paper investigates the maximum size of $r$-wise $t$-intersecting families of $k$-subsets, proving the full $t$-star is largest under certain conditions, improving previous bounds for $r \\geq 3$.

Contribution

The authors establish new bounds on $n$ ensuring the full $t$-star is the largest $r$-wise $t$-intersecting family for $r \\geq 3$, refining prior results.

Findings

Proved the full $t$-star is largest for $n \\geq (2.5t)^{1/(r-1)}(k-t)+k$ when $r \\geq 3$.

Showed the exponent $1/(r-1)$ is optimal through examples.

Improved bounds from recent literature on intersecting families.

Abstract

We consider families, $\mathcal{F}$ of $k$-subsets of an $n$-set. For integers $r\geq 2$, $t\geq 1$, $\mathcal{F}$ is called $r$-wise $t$-intersecting if any $r$ of its members have at least $t$ elements in common. The most natural construction of such a family is the full $t$-star, consisting of all $k$-sets containing a fixed $t$-set. In the case $r=2$ the Exact Erd\H{o}s-Ko-Rado Theorem shows that the full $t$-star is largest if $n\geq (t+1)(k-t+1)$. In the present paper, we prove that for $n\geq (2.5t)^{1/(r-1)}(k-t)+k$, the full $t$-star is largest in case of $r\geq 3$. Examples show that the exponent $\frac{1}{r-1}$ is best possible. This represents a considerable improvement on a recent result of Balogh and Linz.