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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On collections of subsets containing no $ 4$-member Boolean algebra.


Authors: Paul Erdős and Daniel Kleitman
Journal: Proc. Amer. Math. Soc. 28 (1971), 87-90
MSC: Primary 05.04
MathSciNet review: 0270924
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Abstract: In this paper, upper and lower bounds each of the form $ c{2^n}/{n^{1/4}}$ are obtained for the maximum possible size of a collection $ Q$ of subsets of an $ n$ element set satisfying the restriction that no four distinct members $ A,B,C,D$ of $ Q$ satisfy $ A \bigcup B = C$ and $ A \bigcap B = D$.

The lower bound is obtained by a construction while the upper bound is obtained by applying a somewhat weaker condition on $ Q$ which leads easily to a bound. Probably there is an absolute constant $ c$ so that

$\displaystyle \max \vert Q\vert = c{2^n}/{n^{1/4}} + o({2^n}/{n^{1/4}})$

but we cannot prove this and have no guess at what the value of $ c$ is.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1971-0270924-9
PII: S 0002-9939(1971)0270924-9
Keywords: Bounds on collection size, sizes of subset families
Article copyright: © Copyright 1971 American Mathematical Society