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?
Combinatorics Probability and Computing
Bollobás, B., & Eccles, T. (2015). Partial Shadows of Set Systems. Combinatorics Probability and Computing, 24 (5), 825-828. https://doi.org/10.1017/S0963548314000790