Supersaturation in the Boolean lattice
Combinatorics
2017-07-19 v1 Discrete Mathematics
Abstract
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).
Keywords
Cite
@article{arxiv.1303.4336,
title = {Supersaturation in the Boolean lattice},
author = {Andrew P. Dove and Jerrold R. Griggs and Ross J. Kang and Jean-Sébastien Sereni},
journal= {arXiv preprint arXiv:1303.4336},
year = {2017}
}