### Abstract

Let q be a power of a prime p, and let n, d, ℓ be integers such that 1 ≤n, 1 ≤ℓ < q. Consider the modulo q complete ℓ-wide family: ℱ = {F ⊆ [n] : ∃ f ∈ ℤ s.t. d ≤f < d + ℓ and |F| ≡ f (mod q)}. We describe a Grbner basis of the vanishing ideal I(ℱ) of the set of characteristic vectors of ℱ over fields of characteristic p. It turns out that this set of polynomials is a Grbner basis for all term orderings ≺, for which the order of the variables is x _{n} ≺ x_{n-1} ≺ ⋯ ≺ x1. We compute the Hilbert function of I(ℱ), which yields formulae for the modulo p rank of certain inclusion matrices related to ℱ. We apply our results to problems from extremal set theory. We prove a sharp upper bound of the cardinality of a modulo q ℓ-wide family, which shatters only small sets. This is closely related to a conjecture of Frankl [13] on certain ℓ-antichains. The formula of the Hilbert function also allows us to obtain an upper bound on the size of a set system with certain restricted intersections, generalizing a bound proposed by Babai and Frankl [6]. The paper generalizes and extends the results of [15], [16] and [17].

Original language | English |
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Pages (from-to) | 309-333 |

Number of pages | 25 |

Journal | Combinatorics Probability and Computing |

Volume | 18 |

Issue number | 3 |

DOIs | |

Publication status | Published - May 1 2009 |

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### ASJC Scopus subject areas

- Theoretical Computer Science
- Statistics and Probability
- Computational Theory and Mathematics
- Applied Mathematics

### Cite this

*Combinatorics Probability and Computing*,

*18*(3), 309-333. https://doi.org/10.1017/S0963548308009619