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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On collections of subsets containing no $4$-member Boolean algebra.
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by Paul Erdős and Daniel Kleitman PDF
Proc. Amer. Math. Soc. 28 (1971), 87-90 Request permission

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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Additional Information
  • © Copyright 1971 American Mathematical Society
  • 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