### 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) |
---|---|

Pages (from-to) | 219-228 |

Number of pages | 10 |

Journal | Journal of Combinatorial Theory. Series A |

Volume | 73 |

Issue number | 2 |

State | Published - 1996 |

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### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

*Journal of Combinatorial Theory. Series A*,

*73*(2), 219-228.

**On the order dimension of 1-sets versus k-sets.** / Kierstead, Henry.

Research output: Contribution to journal › Article

*Journal of Combinatorial Theory. Series A*, vol. 73, no. 2, pp. 219-228.

}

TY - JOUR

T1 - On the order dimension of 1-sets versus k-sets

AU - Kierstead, Henry

PY - 1996

Y1 - 1996

N2 - 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 Bn(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(log3 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 (log2 n/4 < dim(1, log n; n) 3 n.

AB - 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 Bn(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(log3 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 (log2 n/4 < dim(1, log n; n) 3 n.

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UR - http://www.scopus.com/inward/citedby.url?scp=0042479214&partnerID=8YFLogxK

M3 - Article

VL - 73

SP - 219

EP - 228

JO - Journal of Combinatorial Theory - Series A

JF - Journal of Combinatorial Theory - Series A

SN - 0097-3165

IS - 2

ER -