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A family of sets is called union-closed if whenever A and B are sets of the family, so is A ∪ B. The long-standing union-closed conjecture states that if a family of subsets of [n] is union-closed, some element appears in at least half the sets of the family. A natural weakening is that the union-closed conjecture holds for large families, that is, families consisting of at least p02n sets for some constant p0. The first result in this direction appears in a recent paper of Balla, Bollobás and Eccles , who showed that union-closed families of at least
2n sets satisfy the conjecture; they proved this by determining the minimum possible average size of a set in a union-closed family of given size. However, the methods used in that paper cannot prove a better constant than
. Here, we provide a stability result for the main theorem of , and as a consequence we prove the union-closed conjecture for families of at least (
− c)2n sets, for a positive constant c.
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
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
, which contains most sets in the shadow of
. For example, if
is a family of sets containing all but one set in the shadow of each set of
, how large must