Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
8903767 | Journal of Combinatorial Theory, Series A | 2018 | 17 Pages |
Abstract
We say a family of sets is intersecting if any two of its sets intersect, and we say it is trivially intersecting if there is an element which appears in every set of the family. In this paper we study the maximum size of a non-trivially intersecting family in a natural “multi-part” setting. Here the ground set is divided into parts, and one considers families of sets whose intersection with each part is of a prescribed size. Our work is motivated by classical results in the single-part setting due to ErdÅs, Ko and Rado, and Hilton and Milner, and by a theorem of Frankl concerning intersecting families in this multi-part setting. In the case where the part sizes are sufficiently large we determine the maximum size of a non-trivially intersecting multi-part family, disproving a conjecture of Alon and Katona.
Related Topics
Physical Sciences and Engineering
Mathematics
Discrete Mathematics and Combinatorics
Authors
Matthew Kwan, Benny Sudakov, Pedro Vieira,