Let be an -element set, where is even. We refute a conjecture of J. Gordon and Y. Teplitskaya, according to which, for every maximal intersecting family of -element subsets of , one can partition into disjoint pairs in such a way that no matter how we pick one element from each of the first 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 , we call a family of sets -separable if there is a -element subset such that for every ordered pair of elements of , there exists such that . For a fixed , , and , we establish asymptotically tight estimates for the smallest integer such that every family with is -separable.
"Shattered matchings in intersecting hypergraphs." Mosc. J. Comb. Number Theory 10 (1) 49 - 59, 2021. https://doi.org/10.2140/moscow.2021.10.49