### Abstract

Letχ_{1}(n) denote the maximum possible absolute value of an entry of the inverse of annbyninvertible matrix with 0,1 entries. It is proved thatχ_{1}(n)=n^{(1/2+o(1))n}. This solves a problem of Graham and Sloane. Letm(n) denote the maximum possible numbermsuch that given a set ofmcoins out of a collection of coins of two unknown distinct weights, one can decide if all the coins have the same weight or not usingnweighings in a regular balance beam. It is shown thatm(n)=n^{(1/2+o(1))n}. This settles a problem of Kozlov and Vũ. LetD(n) denote the maximum possible degree of a regular multi-hypergraph onnvertices that contains no proper regular nonempty subhypergraph. It is shown thatD(n)=n^{(1/2+o(1))n}. This improves estimates of Shapley, van Lint and Pollak. All these results and several related ones are proved by a similar technique whose main ingredient is an extension of a construction of Håstad of threshold gates that require large weights.

Original language | English (US) |
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Pages (from-to) | 133-160 |

Number of pages | 28 |

Journal | Journal of Combinatorial Theory. Series A |

Volume | 79 |

Issue number | 1 |

DOIs | |

State | Published - Jul 1 1997 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

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