Uniform intersecting families with large covering number

TL;DR

This paper investigates the maximum size of intersecting families of k-element subsets with a given covering number, revealing surprising behaviors for certain ranges of n, k, and τ, especially when n is less than k squared.

Contribution

It provides new bounds and asymptotic behaviors of M(n,k,τ) for n<k^2 and large τ, extending classical results to these regimes.

Findings

For n = ⌊k^{3/2}⌋ and τ ≤ k - k^{3/4+o(1)}, M(n,k,τ) ≈ (1-o(1)){n-1 choose k-1}

For n = ⌊k^{3/2}⌋ and τ > 0.5k^{1/2}, M(n,k,τ) is exponentially smaller than {n choose k}

The results show unexpected phase transitions in the size of intersecting families depending on n, k, and τ.

Abstract

A family $\mathcal F$ has covering number $\tau$ if the size of the smallest set intersecting all sets from $\mathcal F$ is equal to $\tau$. Let $M(n,k,\tau)$ stand for the size of the largest intersecting family $\mathcal F$ of $k$-element subsets of $\{1,\ldots,n\}$ with covering number $\tau$. It is a classical result of Erd\H os and Lov\'asz that $M(n,k,k)\le k^k$ for any $n$. In this short note, we explore the behaviour of $M(n,k,\tau)$ for $n<k^2$ and large $\tau$. The results are quite surprising: For example, we show that $M(n,k,\tau) =(1-o(1)){n-1\choose k-1}$, if $n = \lfloor k^{3/2}\rfloor$, and $\tau\le k-k^{3/4+o(1)}$ as $k\to\infty$; $M(n,k,\tau) <e^{-ck^{1/2}}{n\choose k}$, if $n = \lfloor k^{3/2}\rfloor$ and $\tau>k-\frac 12k^{1/2}$.