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Standard monomials for q-uniform families and a conjecture of Babai and Frankl

100%

Hegedűs G., Rónyai L.

Open Mathematics

2003

tom 1

nr 2

198-207

Let n, k, α be integers, n, α>0, p be a prime and q=p α. Consider the complete q-uniform family $$\mathcal{F}\left( {k,q} \right) = \left\{ {K \subseteq \left[ n \right]:\left| K \right| \equiv k(mod q)} \right\}$$ We study certain inclusion matrices attached to F(k,q) over the field $$\mathbb{F}_p $$ . We show that if l≤q−1 and 2l≤n then $$rank_{\mathbb{F}_p } I(\mathcal{F}(k,q),\left( {\begin{array}{*{20}c} {\left[ n \right]} \\ { \leqslant \ell } \\ \end{array} } \right)) \leqslant \left( {\begin{array}{*{20}c} n \\ \ell \\ \end{array} } \right)$$ This extends a theorem of Frankl [7] obtained for the case α=1. In the proof we use arguments involving Gröbner bases, standard monomials and reduction. As an application, we solve a problem of Babai and Frankl related to the size of some L-intersecting families modulo q.

Ograniczanie wyników

1 Open Mathematics

1 Hegedűs G.

1 Rónyai L.

1 2003

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Standard monomials for q-uniform families and a conjecture of Babai and Frankl