Let be a union-closed set, then the union-closed set conjecture states that an element exists which belongs to at least of the sets in . Sarvate and Renaud (1989) showed that the conjecture is true if , where is the smallest set in , or if . They also showed that if the conjecture fails, then , where is the largest set of .
These results have since been improved for up to 18 (Sarvate and Renaud 1990), 24 (Lo Faro 1994a), 27 (Poonen 1992), 32 in (Gao and Yu 1998), and the best known result of 40 (Roberts 1992).
The proof for the case where has a 2-set can be effected as follows. Write , then partition the sets of into four disjoint families , , , and , according to whether their intersection with is , , , or , respectively. It follows that by taking unions with , where is the cardinal number of . Now compare with . If , then , so is in at least half the sets of . Similarly, if , then is in at least half the sets (Hoey, pers. comm.).
Unfortunately, this method of proof does not extend to , since Sarvate and Renaud show an example of a union-closed set with where none of , , is in half the sets. However, in these cases, there are other elements which do appear in half the sets, so this is not a counterexample to the conjecture, but only a limitation to the method of proof given above (Hoey, pers. comm.).