Haoran Luo (UIUC)
Maximal 3-wise intersecting families with minimum size: the odd case
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Abstract: A family F on ground set {1,2,...,n} is maximal k-wise intersecting if every collection of k sets in F has a non-empty intersection, and no other set can be added to F while maintaining this property. Erdős and Kleitman asked for the minimum size of a maximal k-wise intersecting family. Complementing earlier work of Hendrey, Lund, Tompkins, and Tran, who answered this question for k=3 and large even n, we answer it for k=3 and large odd n. We show that the unique minimum family is obtained by partitioning the ground set into two sets A and B with sizes differing by at most one and taking the family consisting of all the proper supersets of A and of B. A key ingredient of our proof is the stability result by Ellis and Sudakov about the so-called 2-generator set systems.
This is joint work with József Balogh and Ce Chen.