frankl’s conjecture
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10.37236/6248 ◽  
2017 ◽  
Vol 24 (3) ◽  
Author(s):  
Alireza Abdollahi ◽  
Russ Woodroofe ◽  
Gjergji Zaimi

We show that the subgroup lattice of any finite group satisfies Frankl’s Union-Closed Conjecture. We show the same for all lattices with a modular coatom, a family which includes all supersolvable and dually semimodular lattices. A common technical result used to prove both may be of some independent interest.


ELEMENTOS ◽  
2013 ◽  
Vol 1 (1) ◽  
Author(s):  
Pierre Colomb ◽  
Alexis Irlande ◽  
Olivier Raynaud ◽  
Yoan Renaud

Frankl’s conjecture states that in a family of sets closed by union F such that F 6= {∅}, there is an element that belongs to at least half of the sets of F. There are several partial results of this conjecture. For example, it has been shown that families in which the smallest set is of size 1 or 2, or families closed both by union and by intersection are Frankl’s. In this article, by basing ourselves on an unseen recursive definition of the family of sets closed by union, we will define a new class of Frankl’s families. Subsequently, we will evaluate the size of this class for the first 6 values of n. Finally we will show that this class does not coincide with the already known Frankl’s classes.


Author(s):  
Filip Marić ◽  
Miodrag Živković ◽  
Bojan Vučković

2009 ◽  
Vol 25 (1) ◽  
pp. 115-121 ◽  
Author(s):  
R. S. Shewale ◽  
Vinayak Joshi ◽  
V. S. Kharat

Order ◽  
2009 ◽  
Vol 26 (1) ◽  
pp. 31-48 ◽  
Author(s):  
Gábor Czédli ◽  
Miklós Maróti ◽  
E. Tamás Schmidt

10.37236/812 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Ivica Bošnjak ◽  
Petar Marković

In 1979, P. Frankl conjectured that in a finite union-closed family ${\cal F}$ of finite sets, ${\cal F}\neq\{\emptyset\}$, there has to be an element that belongs to at least half of the sets in ${\cal F}$. We prove this when $|\bigcup{\cal F}|\leq 11$.


2007 ◽  
pp. 29-43 ◽  
Author(s):  
Petar Markovic

In 1979 Frankl conjectured that in a finite union-closed family F of finite sets, F _= {?} there has to be an element that belongs to at least half of the sets in F. We prove this when |U F| _ <10.


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