Long Symmetric Chains in the Boolean Lattice

  title={Long Symmetric Chains in the Boolean Lattice},
  author={B{\'e}la Bajnok and Shahriar Shahriari},
  journal={J. Comb. Theory, Ser. A},
Let 2n]be the poset of all subsets of a set with n elements ordered by inclusion. A long chain in this poset is a chain ofn?1 subsets starting with a subset with one element and ending with a subset withn?1 elements. In this paper we prove: Given any collection of at mostn?2 skipless chains in 2n], there exists at least one (but sometimes not more than one) long chain disjoint from the chains in the collection. Furthermore, fork?3, given a collection ofn?kskipless chains in 2n], there are at… 
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