### Abstract

Let B be any family of 3-subsets of [n] = {1, ..., n} such that every i in [n] belongs to at most three members of B. It is shown here that there exists a 3 × n (0, 1)-matrix M such that every set of columns of M indexed by a member of B is linearly independent over GF(2). The proof depends on finding a suitable vertex-coloring for the associated 3-uniform hypergraph. This matrix result, which is a special case of a conjecture of Griggs and Walker, implies the corresponding special case of a conjecture of Chung, Frankl, Graham, and Shearer and of Faudree, Schelp, and Sós concerning intersecting families of subsets.

Original language | English |
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Pages (from-to) | 306-315 |

Number of pages | 10 |

Journal | Journal of Combinatorial Theory, Series A |

Volume | 53 |

Issue number | 2 |

DOIs | |

Publication status | Published - 1990 |

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### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

*Journal of Combinatorial Theory, Series A*,

*53*(2), 306-315. https://doi.org/10.1016/0097-3165(90)90062-2

**Representations of families of triples over GF(2).** / Füredi, Z.; Griggs, Jerrold R.; Holzman, Ron; Kleitman, Daniel J.

Research output: Contribution to journal › Article

*Journal of Combinatorial Theory, Series A*, vol. 53, no. 2, pp. 306-315. https://doi.org/10.1016/0097-3165(90)90062-2

}

TY - JOUR

T1 - Representations of families of triples over GF(2)

AU - Füredi, Z.

AU - Griggs, Jerrold R.

AU - Holzman, Ron

AU - Kleitman, Daniel J.

PY - 1990

Y1 - 1990

N2 - Let B be any family of 3-subsets of [n] = {1, ..., n} such that every i in [n] belongs to at most three members of B. It is shown here that there exists a 3 × n (0, 1)-matrix M such that every set of columns of M indexed by a member of B is linearly independent over GF(2). The proof depends on finding a suitable vertex-coloring for the associated 3-uniform hypergraph. This matrix result, which is a special case of a conjecture of Griggs and Walker, implies the corresponding special case of a conjecture of Chung, Frankl, Graham, and Shearer and of Faudree, Schelp, and Sós concerning intersecting families of subsets.

AB - Let B be any family of 3-subsets of [n] = {1, ..., n} such that every i in [n] belongs to at most three members of B. It is shown here that there exists a 3 × n (0, 1)-matrix M such that every set of columns of M indexed by a member of B is linearly independent over GF(2). The proof depends on finding a suitable vertex-coloring for the associated 3-uniform hypergraph. This matrix result, which is a special case of a conjecture of Griggs and Walker, implies the corresponding special case of a conjecture of Chung, Frankl, Graham, and Shearer and of Faudree, Schelp, and Sós concerning intersecting families of subsets.

UR - http://www.scopus.com/inward/record.url?scp=38249019319&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=38249019319&partnerID=8YFLogxK

U2 - 10.1016/0097-3165(90)90062-2

DO - 10.1016/0097-3165(90)90062-2

M3 - Article

VL - 53

SP - 306

EP - 315

JO - Journal of Combinatorial Theory - Series A

JF - Journal of Combinatorial Theory - Series A

SN - 0097-3165

IS - 2

ER -