Partial Shadows of Set Systems


The shadow of a system of sets is all sets which can be obtained by taking a set in the original system, and removing a single element. The Kruskal-Katona theorem tells us the minimum possible size of the shadow of A, if A consists of m r-element sets. In this paper, we ask questions and make conjectures about the minimum possible size of a partial shadow for A, which contains most sets in the shadow of A. For example, if B is a family of sets containing all but one set in the shadow of each set of A, how large must be?

Publication Title

Combinatorics Probability and Computing