Union-closed families

AbstractA union-closed family F is a finite collection of sets not all empty, such that any union of elements of F is itself an element of F. Peter Frankl conjectured in 1979 that for any such family, there is an element in at least half of its sets. But the problem remains unsolved. We find a number of equivalent conjectures, and we prove the conjecture in special cases, including for example all families involving up to seven elements or having up to 28 sets, extending the previously known result for up to 18 sets. We also prove a general theorem stating exactly when a subfamily is enough to guarantee the existence of an element from the subfamily which is in half the sets of the whole family