Union-Closed Sets Conjecture
Let ![A=<span style=]()
{A_1,A_2,...,A_n}" src="https://mathworld.wolfram.com/images/equations/Union-ClosedSetsConjecture/Inline1.gif" style="height:15px; width:119px" /> 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 ![A_1=<span style=]()
{x,y}" src="https://mathworld.wolfram.com/images/equations/Union-ClosedSetsConjecture/Inline13.gif" style="height:15px; width:64px" />, then partition the sets of 
into four disjoint families 
, 
, 
, and 
, according to whether their intersection with 
is 
, ![<span style=]()
{x}" src="https://mathworld.wolfram.com/images/equations/Union-ClosedSetsConjecture/Inline21.gif" style="height:15px; width:17px" />, ![<span style=]()
{y}" src="https://mathworld.wolfram.com/images/equations/Union-ClosedSetsConjecture/Inline22.gif" style="height:15px; width:17px" />, or ![<span style=]()
{x,y}" src="https://mathworld.wolfram.com/images/equations/Union-ClosedSetsConjecture/Inline23.gif" style="height:15px; width:32px" />, 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 ![A_1=<span style=]()
{x,y,z}" src="https://mathworld.wolfram.com/images/equations/Union-ClosedSetsConjecture/Inline37.gif" style="height:15px; width:78px" /> 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.).
REFERENCES:
Gao, W. and Yu, H. "Note on the Union-Closed Sets Conjecture." Ars Combin. 49, 280-288, 1998.
Lo Faro, G. "A Note on the Union-Closed Sets Conjecture." J. Austral. Math. Soc. Ser. A 57, 230-236, 1994a.
Lo Faro, G. "Union-Closed Sets Conjecture: Improved Bounds." J. Combin. Math. Combin. Comput. 16, 97-102, 1994b.
Poonen, B. "Union-Closed Families." J. Combin. Theory Ser. A 59, 253-268, 1992.
Roberts, I. Tech. Rep. No. 2/92. School Math. Stat., Curtin Univ. Tech., Perth, 1992.
Sarvate, D. G. and Renaud, J.-C. "On the Union-Closed Sets Conjecture." Ars Combin. 27, 149-153, 1989.
Sarvate, D. G. and Renaud, J.-C. "Improved Bounds for the Union-Closed Sets Conjecture." Ars Combin. 29, 181-185, 1990.
West, D. "Union-Closed Sets Conjecture (1979)." http://www.math.uiuc.edu/~west/openp/unionclos.html.
