Uniform intersecting families with large covering number
Peter Frankl, Andrey Kupavskii

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 has covering number if the size of the smallest set intersecting all sets from is equal to . Let stand for the size of the largest intersecting family of -element subsets of with covering number . It is a classical result of Erd\H os and Lov\'asz that for any . In this short note, we explore the behaviour of for and large . The results are quite surprising: For example, we show that , if , and as ; , if and .
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Taxonomy
TopicsLimits and Structures in Graph Theory · Advanced Graph Theory Research · Analytic Number Theory Research
