### Abstract

The general problem of dividing a set into a maximal collection of subsets in such a way that the subsets will overlap in certain specified ways is a fundamental problem in experimental design and in the design of systems involving shared use of system elements. This memorandum solves this problem in the case for which the subsets are each required to have three elements, and any pair of subsets have at most one element in common. A family F of three element subsets of an n-element set S_{n} is called n-consistent if the intersection of any two sets of F contain at most one element of S_{n}. We find maximal (in number of elements) F for all n. For certain n the F are Steiner Triples Systems. The construction of the F is constructive. Structure Theorems are given determining the graph of doublets not covered by triplets in F.

Original language | English (US) |
---|---|

Pages (from-to) | 1-8 |

Number of pages | 8 |

Journal | Journal of Combinatorial Theory |

Volume | 5 |

Issue number | 1 |

State | Published - Jul 1968 |

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### Cite this

*Journal of Combinatorial Theory*,

*5*(1), 1-8.

**Maximal consistent families of triples.** / Spencer, Joel.

Research output: Contribution to journal › Article

*Journal of Combinatorial Theory*, vol. 5, no. 1, pp. 1-8.

}

TY - JOUR

T1 - Maximal consistent families of triples

AU - Spencer, Joel

PY - 1968/7

Y1 - 1968/7

N2 - The general problem of dividing a set into a maximal collection of subsets in such a way that the subsets will overlap in certain specified ways is a fundamental problem in experimental design and in the design of systems involving shared use of system elements. This memorandum solves this problem in the case for which the subsets are each required to have three elements, and any pair of subsets have at most one element in common. A family F of three element subsets of an n-element set Sn is called n-consistent if the intersection of any two sets of F contain at most one element of Sn. We find maximal (in number of elements) F for all n. For certain n the F are Steiner Triples Systems. The construction of the F is constructive. Structure Theorems are given determining the graph of doublets not covered by triplets in F.

AB - The general problem of dividing a set into a maximal collection of subsets in such a way that the subsets will overlap in certain specified ways is a fundamental problem in experimental design and in the design of systems involving shared use of system elements. This memorandum solves this problem in the case for which the subsets are each required to have three elements, and any pair of subsets have at most one element in common. A family F of three element subsets of an n-element set Sn is called n-consistent if the intersection of any two sets of F contain at most one element of Sn. We find maximal (in number of elements) F for all n. For certain n the F are Steiner Triples Systems. The construction of the F is constructive. Structure Theorems are given determining the graph of doublets not covered by triplets in F.

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UR - http://www.scopus.com/inward/citedby.url?scp=0039313605&partnerID=8YFLogxK

M3 - Article

VL - 5

SP - 1

EP - 8

JO - Journal of Combinatorial Theory

JF - Journal of Combinatorial Theory

SN - 0021-9800

IS - 1

ER -