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On Kruskal's cascades and counting containments in a set of subsets

Part of: Combinatorics

Published online by Cambridge University Press:  26 February 2010

David E. Daykin  and
Peter Frankl
David E. Daykin
Affiliation:
Dept. of Mathematics, The University of Reading, Whiteknights, Reading RG6 2AX
Peter Frankl
Affiliation:
C.N.R.S., Paris, France
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Abstract

Let ℱ be a set of m subsets of X = {1,2,…, n}. We study the maximum number λ of containments Y ⊂ Z with Y, Z ∊ ℱ. Theorem 9. , if, and only if, ml/n → 1. When n is large and members of ℱ have cardinality k or k–1 we determine λ. For this we bound (ΔN)/N where ΔN is the shadow of Kruskal's k-cascade for the integer N. Roughly, if m ∼ N + ΔN, then λ ∼ kN with infinitely many cases of equality. A by-product is Theorem 7 of LYM posets.

MSC classification

Secondary: 05A05: Permutations, words, matrices

Information

Type
Research Article
Information
Mathematika , Volume 30 , Issue 1 , June 1983 , pp. 133 - 141
Copyright
Copyright © University College London 1983

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