Supersaturation in the Boolean lattice

Supersaturation dans le treillis booléen

Andrew P. Dove (1) , Jerrold R. Griggs (1) , Ross J. Kang (2) , Jean-Sébastien Sereni (3)

1 Department of Mathematics [Columbia]
2 Mathematical Institute
3 ORPAILLEUR - Knowledge representation, reasonning

We prove a "supersaturation-type'' extension of both Sperner's Theorem (1928) and its generalization by Erdos (1945) to k-chains. Our result implies that a largest family whose size is x more than the size of a largest k-chain free family and that contains the minimum number of k-chains is the family formed by taking the middle (k-1) rows of the Boolean lattice and x elements from the k-th middle row. We prove our result using the symmetric chain decomposition method of de Bruijn, van Ebbenhorst Tengbergen, and Kruyswijk (1951).

Domaines: Combinatoire [math.CO]; Mathématique discrète [cs.DM]

Andrew P. Dove, Jerrold R. Griggs, Ross J. Kang, Jean-Sébastien Sereni. Supersaturation in the Boolean lattice. Integers : Electronic Journal of Combinatorial Number Theory, 2014, The Dick de Bruijn Memorial Volume, 14A, pp.A4. ⟨hal-00802000⟩