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The proof of a conjecture of G. O. H. Katona

Authors
Journal
Journal of Combinatorial Theory Series A
0097-3165
Publisher
Elsevier
Publication Date
Volume
19
Issue
2
Identifiers
DOI: 10.1016/s0097-3165(75)80009-4

Abstract

The following conjecture of G. O. H. Katona is proved. Let X be a finite set of cardinality n, and A a family of subsets of X. Let us suppose that for any two members A, B of A we have | A ∪ B|≤ n-r, | A ∩ B| ≥1, r is a positive integer, r≤n. Then | A | ⩽ ∑ i = 0 ( n - 1 - r ) / 2 ( n − 1 i ) for odd, and | A | ⩽ ∑ i = 0 ( n - 2 - r ) / 2 ( n − 1 i ) + ( n − 2 ( n − 2 − r ) / 2 ) for even values of n-r.

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