Uniform intersecting families with large covering number

2021/06/09 by Péter Frankl, Frankl, Peter, Andrey Kupavskii +1 · 1 citation
Mathematics · Computer Science · #Limits and Structures in Graph Theory #Advanced Graph Theory Research #Analytic Number Theory Research

paper · doi:10.48550/arxiv.2106.05344

Abstract

A family \mathcal F has covering number τ if the size of the smallest set intersecting all sets from \mathcal F is equal to τ. Let M(n,k,τ) stand for the size of the largest intersecting family \mathcal F of k-element subsets of \1,…,n\ with covering number τ. It is a classical result of Erd\H os and Lovász that M(n,k,k)≤ k^k for any n. In this short note, we explore the behaviour of M(n,k,τ) for nk-\frac 12k^1/2.

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