Details:
Title  Balancing sets of vectors.  Author(s)  Gleb Gusev  Type  Article in Journal  Abstract  Let n be an arbitrary integer, let p be a prime factor of n . Denote by ω 1 the p th primitive unity root,
.Define ω i ≔ ω 1 i for 0 ≦ i ≦ p − 1 and B ≔ {1, ω 1 , …, ω p −1 } n ⊆ ℂ n .Denote by K ( n; p ) the minimum k for which there exist vectors ν 1 , …, ν k ∈ B such that for any vector w ∈ B , there is an i , 1 ≦ i ≦ k , such that ν i · w = 0, where ν · w is the usual scalar product of ν and w .Gröbner basis methods and linear algebra proof gives the lower bound K ( n; p ) ≧ n ( p − 1).Galvin posed the following problem: Let m = m ( n ) denote the minimal integer such that there exists subsets A 1 , …, A m of {1, …, 4 n } with  A i  = 2 n for each 1 ≦ i ≦ n , such that for any subset B ⊆ [4 n ] with 2 n elements there is at least one i , 1 ≦ i ≦ m , with A i ∩ B having n elements. We obtain here the result m ( p ) ≧ p in the case of p > 3 primes.  Keywords  Gröbner bases, polynomial ideals, standard monomials, extremal combinatorics  ISSN  00816906; 15882896/e 
URL 
http://www.akademiai.com/doi/abs/10.1556/SScMath.2009.1134 
Language  English  Journal  Stud. Sci. Math. Hung.  Volume  47  Number  3  Pages  333349  Publisher  Akad  Year  2010  Edition  0  Translation 
No  Refereed 
No 
