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

Erdős, Paul; Kleitman, Daniel

doi:10.1090/S0002-9939-1971-0270924-9

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.

References

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Problem

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A problem of Zarankiewicz

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