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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
DOI: https://doi.org/10.1090/S0002-9939-1971-0270924-9
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 \[ \max |Q| = c{2^n}/{n^{1/4}} + o({2^n}/{n^{1/4}})\] but we cannot prove this and have no guess at what the value of $c$ is.


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Keywords: Bounds on collection size, sizes of subset families
Article copyright: © Copyright 1971 American Mathematical Society