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.
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Ahlswede, R., Cai, N. (2013). Cross-Disjoint Pairs of Clouds in the Interval Lattice. In: Graham, R., Nešetřil, J., Butler, S. (eds) The Mathematics of Paul Erdős I. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-7258-2_8
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