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

Gábor Hegedűs, and Lajos Rónyai. "Standard monomials for q-uniform families and a conjecture of Babai and Frankl." Open Mathematics 1.2 (2003): 198-207. <http://eudml.org/doc/268782>.

@article{GáborHegedűs2003,
abstract = {Let n, k, α be integers, n, α>0, p be a prime and q=p α. Consider the complete q-uniform family \[\mathcal \{F\}\left( \{k,q\} \right) = \left\lbrace \{K \subseteq \left[ n \right]:\left| K \right| \equiv k(mod q)\} \right\rbrace \] 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.},
author = {Gábor Hegedűs, Lajos Rónyai},
journal = {Open Mathematics},
keywords = {05D05; 13P10; 05B20},
language = {eng},
number = {2},
pages = {198-207},
title = {Standard monomials for q-uniform families and a conjecture of Babai and Frankl},
url = {http://eudml.org/doc/268782},
volume = {1},
year = {2003},
}

TY - JOUR
AU - Gábor Hegedűs
AU - Lajos Rónyai
TI - Standard monomials for q-uniform families and a conjecture of Babai and Frankl
JO - Open Mathematics
PY - 2003
VL - 1
IS - 2
SP - 198
EP - 207
AB - Let n, k, α be integers, n, α>0, p be a prime and q=p α. Consider the complete q-uniform family \[\mathcal {F}\left( {k,q} \right) = \left\lbrace {K \subseteq \left[ n \right]:\left| K \right| \equiv k(mod q)} \right\rbrace \] 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.
LA - eng
KW - 05D05; 13P10; 05B20
UR - http://eudml.org/doc/268782
ER -