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On A Problem of Purdy Related to Sperner Systems

Published online by Cambridge University Press:  20 November 2018

J. Schonheim
J. Schonheim*
Affiliation:
Tel Aviv University, Tel Aviv, Israel
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Purdy asked whether the following conjecture is true

Conjecture. Let E be a set of 2n elements. If S={Sl, S2, …, St} is a Sperner system of E, i.e. for i≠j, i, j, =1, 2, …, t; and if

(1)

then

The proof of the conjecture will be obtained using the following theorem of Katona (Acta Math. 15 (1964), 329-337):

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 17 , Issue 1 , 01 March 1974 , pp. 135 - 136
Copyright
Copyright © Canadian Mathematical Society 1974

References

1. De Bruijn, N. G. and al. On the set of divisors of a number, Nieuw Arch. Wisk. 23 (1952), 191-193.Google Scholar
2. Erdös, P. and al. Intersection theorems for systems of finite sets, Quart. J. Math. Oxford Ser. 12 (1961), 313-320.Google Scholar