### Abstract

A family F of s-subsets of [t] is a (θ, s, t)-family iff the intersection of any two distinct elements of F has cardinality less than θ. Let f(θ, s, t) be the greatest integer n such that there exists an (θ, s, t)-family of cardinality n. Let dim(1, k; n) denote the dimension of B_{n}(1, k), the suborder of the Boolean lattice on [n] consisting of 1-subsets and k-subsets of [n]. We use upper and lower bounds on f(θ, s, t) to derive new lower and upper bounds on dim(1, k; n). In particular we answer a question of Trotter by showing that dim(1, log n; n) = Ω(log^{3} n/log log n). The estimation of dim(1, log n; n) plays a critical role in the determination of the maximum dimension of an ordered set with fixed maximum degree. Previously it was only known that (log^{2} n/4 < dim(1, log n; n) <log^{3} n.

Original language | English (US) |
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Pages (from-to) | 219-228 |

Number of pages | 10 |

Journal | Journal of Combinatorial Theory. Series A |

Volume | 73 |

Issue number | 2 |

DOIs | |

State | Published - Jan 1 1996 |

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### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics