Improved Bounds for the Union-Closed Sets Conjecture

It has been conjectured that for any union-closed set \(A\) there exists some element which is contained in at least half the sets in \(A\).
This has recently been shown that this conjecture hold if the smallest set in \(A\) has size one or two, and also to hold if the number of sets in \(A\) is less than eleven.It is shown that the smallest set size approach is unproductive for size three. It is also shown that the conjecture holds for other conditions on the sets in \(A\), and an improved bound is derived: the conjecture holds if the number of sets in \(A\) is less than 19.