Abstract
Theshadowof 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$\mathcal A$, if$\mathcal A$consists ofm r-element sets.In this paper, we ask questions and make conjectures about the minimum possible size of apartial shadowfor$\mathcal A$, which contains most sets in the shadow of$\mathcal A$. For example, if$\mathcal B$is a family of sets containing all but one set in the shadow of each set of$\mathcal A$, how large must$\mathcal B$be?
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