Algebraic properties of modulo q complete ℓ-wide families

  • Authors:

  • BÁlint Felszeghy;GÁbor HegedŰs;Lajos RÓnyai

  • Affiliations:

  • Institute of mathematics, budapest university of technology and economics, budapest, hungary (e-mail: fbalint@math.bme.hu);Kecskemét, hungary (e-mail: greece@math.bme.hu);Computer and automation institute, hungarian academy of sciences, budapest, hungary and institute of mathematics, budapest university of technology and economics, budapest, hungary (e-mail: lajos@ ...

  • Venue:
  • Combinatorics, Probability and Computing
  • Year:
  • 2009

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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: \[ \cF = \bigl\{F\subseteq[n]\,:\, \exists\,f\in\Z~~\text{s.t.}~~ d\leq f We describe a Gröbner basis of the vanishing ideal I() of the set of characteristic vectors of over fields of characteristic p. It turns out that this set of polynomials is a Gröbner basis for all term orderings ≺, for which the order of the variables is xn ≺ xn−1 ≺ ⋅⋅⋅ ≺ x1. We compute the Hilbert function of I(), which yields formulae for the modulo p rank of certain inclusion matrices related to . 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].