Cross-Disjoint Pairs of Clouds in the Interval Lattice

Rudolf Ahlswede and Ning Cai ()
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Rudolf Ahlswede: Universität Bielefeld, Fakultät für Mathematik
Ning Cai: Universität Bielefeld, Fakultät für Mathematik

A chapter in The Mathematics of Paul Erdős I, 2013, pp 107-117 from Springer

Abstract: Summary Let $$\mathcal{I}_{n}$$ be the lattice of intervals in the Boolean lattice $$\mathcal{L}_{n}$$ . For $$\mathcal{A},\mathcal{B}\,\subset \,\mathcal{I}_{n}$$ the pair of clouds $$(\mathcal{A},\mathcal{B})$$ is cross-disjoint, if $$I \cap J = \emptyset $$ for $$I \in \mathcal{A},\ J \in \mathcal{B}$$ . We prove that for such pairs $$\vert \mathcal{A}\vert \vert \mathcal{B}\vert \leq {3}^{2n-2}$$ and that this bound is best possible. Optimal pairs are up to obvious isomorphisms unique. The proof is based on a new bound on cross intersecting families in $$\mathcal{L}_{n}$$ with a weight distribution. It implies also an Intersection Theorem for multisets of Erdős P, Schőnheim J (1969) On the set of non pairwise coprime division of a number. In: Proc. of the Colloquium on Comb. Math. Dalaton Füred, pp 369–376.

Keywords: Latent Interval; Intersection Theorem; Optimal Pair; Sequence Terminology; Bigger Weight (search for similar items in EconPapers)
Date: 2013
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DOI: 10.1007/978-1-4614-7258-2_8

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