Geodesic
Venn diagrams and symmetric chain decompositions in the Boolean lattice
The electronic journal of combinatorics, Tome 11 (2004) no. 1
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We show that symmetric Venn diagrams for $n$ sets exist for every prime $n$, settling an open question. Until this time, $n=11$ was the largest prime for which the existence of such diagrams had been proven, a result of Peter Hamburger. We show that the problem can be reduced to finding a symmetric chain decomposition, satisfying a certain cover property, in a subposet of the Boolean lattice ${\cal B}_n$, and prove that such decompositions exist for all prime $n$. A consequence of the approach is a constructive proof that the quotient poset of ${\cal B}_n$, under the relation "equivalence under rotation", has a symmetric chain decomposition whenever $n$ is prime. We also show how symmetric chain decompositions can be used to construct, for all $n$, monotone Venn diagrams with the minimum number of vertices, giving a simpler existence proof.
DOI : 10.37236/1755
Classification : 06A07, 05C10
Mots-clés : Venn diagrams, symmetric chain decomposition, Boolean lattice 
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     author = {Jerrold Griggs and Charles E. Killian and Carla D. Savage},
     title = {Venn diagrams and symmetric chain decompositions in the {Boolean} lattice},
     journal = {The electronic journal of combinatorics},
     year = {2004},
     volume = {11},
     number = {1},
     doi = {10.37236/1755},
     zbl = {1034.06001},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/1755/}
}
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Jerrold Griggs; Charles E. Killian; Carla D. Savage. Venn diagrams and symmetric chain decompositions in the Boolean lattice. The electronic journal of combinatorics, Tome 11 (2004) no. 1. doi: 10.37236/1755

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