Three layer $Q_2$-free families in the Boolean lattice

Jacob Manske; Jian Shen

Three layer $Q_2$-free families in the Boolean lattice

Combinatorics 2011-08-23 v1

Authors: Jacob Manske , Jian Shen

Abstract

We prove that the largest $Q_2$ -free family of subsets of $[n]$ which contains sets of at most three different sizes has at most $(3 + 2\sqrt {3})N/3 + o(N) \approx 2.1547N + o(N)$ members, where $N = {n \choose {\lfloor n/2 \rfloor}}$ . This improves an earlier bound of $2.207N + o(N)$ by Axenovich, Manske, and Martin.

Keywords

Q_2-free family Boolean lattice extremal set theory forbidden subposet Sperner's theorem

Cite

@article{arxiv.1108.4373,
  title  = {Three layer $Q_2$-free families in the Boolean lattice},
  author = {Jacob Manske and Jian Shen},
  journal= {arXiv preprint arXiv:1108.4373},
  year   = {2011}
}

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