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 p 02n sets for some constant p 0. The first result in this direction appears in a recent paper of Balla, Bollobás and Eccles [1], who showed that union-closed families of at least $\tfrac{2}{3}$ 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 $\tfrac{2}{3}$ . Here, we provide a stability result for the main theorem of [1], and as a consequence we prove the union-closed conjecture for families of at least ( $\tfrac{2}{3}$ − 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 $\mathcal A$ , if $\mathcal A$ 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 $\mathcal A$ , 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?