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Sets of finite sets satisfying union conditions

Published online by Cambridge University Press:  26 February 2010

David E. Daykin  and
Peter Frankl
David E. Daykin
Affiliation:
University of Reading, Reading. RG6 2AX
Peter Frankl
Affiliation:
C.N.R.S., Paris, France.
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Abstarct

Let ℱ denote a set of subsets of X = {1, 2,…, n). Let deg(i) be the number of members of ℱ containing i and val(ℱ) = min {deg (i): i ∈ X). Suppose no k members of ℱ have union X. We conjecture val(ℱ) ≤ 2n-k-1 for k ≥ 3. This is known for n ≤ 2k and we prove it for k ≥ 25. For k = 2 an example has val(ℱ) > 2n-2(l−n-0·651) and we prove val(ℱ) ≤ 2n-2(1–n-1). We also prove that if the union of k sets one from each of ℱ1,…, ℱk has cardinality at most n – t then min {cardinality ℱj} < 2nαt where αk = 2α − 1 and ½ < α < 1.

Type
Research Article
Information
Mathematika , Volume 29 , Issue 1 , June 1982 , pp. 128 - 134
Copyright
Copyright © University College London 1982

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