Standard monomials forq-uniform families and a conjecture of Babai and Frankl

✍ Scribed by Gábor Hegedűs; Lajos Rónyai

• Book ID: 111487711
• Publisher: SP Versita
• Year: 2003
• Tongue: English
• Weight: 224 KB
• Volume: 1
• Category: Article
• ISSN: 1895-1074
• DOI: 10.2478/bf02476008

✦ Synopsis

Let n; k; ¬ be integers, n; ¬ > 0, p be a prime and q = p . Consider the complete q-uniform family F (k; q) = fK [n] : jKj ² k (mod q)g:

We study certain inclusion matrices attached to F (k; q) over the eld F p . We show that if µ q ¡ 1 and 2µ n then rank p I(F (k; q);

This extends a theorem of Frankl [7] obtained for the case ¬ = 1. In the proof we use arguments involving Gr obner 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.