| Title:
|
Frankl’s conjecture for large semimodular and planar semimodular lattices (English) |
| Author:
|
Czédli, Gábor |
| Author:
|
Schmidt, E. Tamás |
| Language:
|
English |
| Journal:
|
Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica |
| ISSN:
|
0231-9721 |
| Volume:
|
47 |
| Issue:
|
1 |
| Year:
|
2008 |
| Pages:
|
47-53 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
A lattice $L$ is said to satisfy (the lattice theoretic version of) Frankl’s conjecture if there is a join-irreducible element $f\in L$ such that at most half of the elements $x$ of $L$ satisfy $f\le x$. Frankl’s conjecture, also called as union-closed sets conjecture, is well-known in combinatorics, and it is equivalent to the statement that every finite lattice satisfies Frankl’s conjecture. Let $m$ denote the number of nonzero join-irreducible elements of $L$. It is well-known that $L$ consists of at most $2^m$ elements. Let us say that $L$ is large if it has more than $5\cdot 2^{m-3}$ elements. It is shown that every large semimodular lattice satisfies Frankl’s conjecture. The second result states that every finite semimodular planar lattice $L$ satisfies Frankl’s conjecture. If, in addition, $L$ has at least four elements and its largest element is join-reducible then there are at least two choices for the above-mentioned $f$. (English) |
| Keyword:
|
union-closed sets |
| Keyword:
|
Frankl’s conjecture |
| Keyword:
|
lattice |
| Keyword:
|
semimodularity |
| Keyword:
|
planar lattice |
| MSC:
|
05A05 |
| MSC:
|
05B35 |
| MSC:
|
06A07 |
| MSC:
|
06E99 |
| idZBL:
|
Zbl 1187.05002 |
| idMR:
|
MR2482716 |
| . |
| Date available:
|
2009-08-27T11:28:57Z |
| Last updated:
|
2012-05-04 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/133405 |
| . |
| Reference:
|
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| Reference:
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| Reference:
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| Reference:
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| Reference:
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| Reference:
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