### Abstract

In this paper, upper and lower bounds each of the form c2^{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 ∪ B = C and A ∩ 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| = c2^{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.

Original language | English |
---|---|

Pages (from-to) | 87-90 |

Number of pages | 4 |

Journal | Proceedings of the American Mathematical Society |

Volume | 28 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1971 |

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### Keywords

- Bounds on collection size
- Sizes of subset families

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Proceedings of the American Mathematical Society*,

*28*(1), 87-90. https://doi.org/10.1090/S0002-9939-1971-0270924-9

**On collections of subsets containing no 4-member boolean algebra.** / Erdős, P.; Kleitman, Daniel.

Research output: Contribution to journal › Article

*Proceedings of the American Mathematical Society*, vol. 28, no. 1, pp. 87-90. https://doi.org/10.1090/S0002-9939-1971-0270924-9

}

TY - JOUR

T1 - On collections of subsets containing no 4-member boolean algebra

AU - Erdős, P.

AU - Kleitman, Daniel

PY - 1971

Y1 - 1971

N2 - In this paper, upper and lower bounds each of the form c2n/n1/4 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 ∪ B = C and A ∩ 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| = c2n/n1/4 + o(2n/n1/4) but we cannot prove this and have no guess at what the value of c is.

AB - In this paper, upper and lower bounds each of the form c2n/n1/4 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 ∪ B = C and A ∩ 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| = c2n/n1/4 + o(2n/n1/4) but we cannot prove this and have no guess at what the value of c is.

KW - Bounds on collection size

KW - Sizes of subset families

UR - http://www.scopus.com/inward/record.url?scp=0347919083&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0347919083&partnerID=8YFLogxK

U2 - 10.1090/S0002-9939-1971-0270924-9

DO - 10.1090/S0002-9939-1971-0270924-9

M3 - Article

VL - 28

SP - 87

EP - 90

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

SN - 0002-9939

IS - 1

ER -