From: mokie@cco.caltech.edu (Michael L. Brundage)
Newsgroups: sci.math.research
Subject: Union-closed Sets
Date: 2 Apr 92 21:38:39 GMT

In my Combinatorics class, we were presented with the open Conjecture by
P. Frankl (sp?):
	Let A be a union-closed family (a non-empty finite collection of
distinct non-empty sets, closed under union).  Then there exists an element
of A which belongs in at least half the sets of A.
	
	Has any progress been made on this?

-- 
mokie@cco.caltech.edu        |  "Let not your heart be troubled,
        or                   |    believe in Christ, believe also in Me."
michaelb@iago.caltech.edu    |       John 14:1 (NASVB)

==============================================================================
From: "Timothy Chow" <tchow@math.lsa.umich.edu>
Date: Mon, 17 Feb 1997 14:59:37 -0500 (EST)
To: rusin@math.niu.edu
Subject: www.math.niu.edu/~rusin/known-math

[deletia]
1. Two references for the Frankl problem are Bjorn Poonen, Union-closed
   families, J. Combin. Theory A59 (1992), 253-268, and Piotr Wojcik,
   Density of union-closed families, Disc. Math. 105 (1992), 259-267.
[deletia]

Tim.
==============================================================================
   Nishimura, Takashi(J-YOKOED); Takahashi, Shin
   Around Frankl conjecture. (English)
   Sci. Rep. Yokohama Nat. Univ. Sect. I Math. Phys. Chem. No. 43 (1996),
   15--23.
     _________________________________________________________________
   
   The subject of the paper is the following conjecture of Frankl: if a
   set system on the underlying set $[n]$ has at least two elements and
   is closed under intersection, then there must be an element $i$ of the
   underlying set which is contained in at most half of the elements of
   the set system.
   
   The authors prove that this conjecture is true if the maximal elements
   of the set system are of size $n-1$ or $n-2$. The conjecture is also
   shown to be true if the size of these maximal elements is not larger
   than $n/2$. Other special cases are considered: the conjecture is
   shown to be true if the set system has more than $2\sp n-2\sp {n-k-1}$
   elements, where $k=[(n-1)/2]$. Finally, it is proved that the
   conjecture is true with the following supplementary condition: if $F$
   is an element of the set system, then all subsets of $F$ whose sizes
   have the same parity as $\vert F\vert $ must also be elements of the
   set system, too.
             Reviewed by Miklos Bona

   © Copyright American Mathematical Society 1997