Intersecting families with covering number three

TL;DR

This paper establishes a precise upper bound on the size of intersecting k-graphs with covering number at least three for sufficiently large n, extending previous results with a new tight bound.

Contribution

It proves a new exact upper bound for the size of such intersecting families when k≥7 and n≥2k, improving upon earlier exponential constraints.

Findings

Derived the best possible upper bound for intersecting k-graphs with covering number ≥3.

Extended previous results to larger classes of k-graphs with tighter bounds.

Confirmed the bound's optimality under specified conditions.

Abstract

We consider $k$-graphs on $n$ vertices, that is, $\mathcal{F}\subset \binom{[n]}{k}$. A $k$-graph $\mathcal{F}$ is called intersecting if $F\cap F'\neq \emptyset$ for all $F,F'\in \mathcal{F}$. In the present paper we prove that for $k\geq 7$, $n\geq 2k$, any intersecting $k$-graph $\mathcal{F}$ with covering number at least three, satisfies $|\mathcal{F}|\leq \binom{n-1}{k-1}-\binom{n-k}{k-1}-\binom{n-k-1}{k-1}+\binom{n-2k}{k-1}+\binom{n-k-2}{k-3}+3$, the best possible upper bound which was proved in \cite{F80} subject to exponential constraints $n>n_0(k)$.