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2021 Shattered matchings in intersecting hypergraphs
Peter Frankl, János Pach
Mosc. J. Comb. Number Theory 10(1): 49-59 (2021). DOI: 10.2140/moscow.2021.10.49

Abstract

Let X be an n-element set, where n is even. We refute a conjecture of J. Gordon and Y. Teplitskaya, according to which, for every maximal intersecting family of n2-element subsets of X, one can partition X into n2 disjoint pairs in such a way that no matter how we pick one element from each of the first n21 pairs, the set formed by them can always be completed to a member of by adding an element of the last pair.

The above problem is related to classical questions in extremal set theory. For any t2, we call a family of sets 2X t-separable if there is a t-element subset TX such that for every ordered pair of elements (x,y) of T, there exists F such that F{x,y}={x}. For a fixed t, 2t5, and n, we establish asymptotically tight estimates for the smallest integer s=s(n,t) such that every family with ||s is t-separable.

Citation

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Peter Frankl. János Pach. "Shattered matchings in intersecting hypergraphs." Mosc. J. Comb. Number Theory 10 (1) 49 - 59, 2021. https://doi.org/10.2140/moscow.2021.10.49

Information

Received: 10 May 2020; Revised: 27 July 2020; Accepted: 12 August 2020; Published: 2021
First available in Project Euclid: 22 January 2021

Digital Object Identifier: 10.2140/moscow.2021.10.49

Subjects:
Primary: 05C65, 05D05, 05D40

Rights: Copyright © 2021 Mathematical Sciences Publishers

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