Standard monomials for <Emphasis Type="Italic">q</Emphasis>-uniform families and a conjecture of Babai and Frankl |
| |
Authors: | Gábor Heged?s Lajos Rónyai |
| |
Institution: | 1.Computer and Automation Institute,Hungarian Academy of Sciences,Budapest,Hungary;2.Budapest University of Technology and Economics,Budapest,Hungary |
| |
Abstract: | 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 2 l≤ 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. |
| |
Keywords: | Gr?bner basis inclusion matrix set family |
本文献已被 SpringerLink 等数据库收录! |
|