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UBC Theses and Dissertations

New results on some Erdős-Ko-Rado-type problems Currier, Gabriel


In this thesis, we study two longstanding problems in extremal set theory related to the Erdős-Ko-Rado theorem. In these problems, we have a collection F of k-subsets of an n-set that contains no copy of some forbidden substructure. The forbidden substructures in question here are known as clusters and simplices and are defined according to intersection and union constraints. The cluster conjecture that we consider was made by Mubayi in 2006 and the simplex conjecture was made in 1974 by Chvátal. We resolve completely the first of these two conjectures, and resolve the second for all but very small values of n.

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