## Summary

Let **F** be a field, and α_{0},...,α_{k-1} be *k* distinct elements of **F**. Let λ =(λ_{1},...,λ_{k}) be a partition of *n* and *V*
_{λ} be the set of all vectors *v*=(*v*
_{1},...,*v*
_{n})∈ **F**
^{n} such that |{*j* ∈ [*n*] : *v*
_{j}=α_{i}}|=λ_{i+1} for 0≦ *i* ≦\ *k-1*. We describe the lexicographic standard monomials of the ideal of polynomials from **F**[*x*
_{1},...,*x*
_{n}] which vanish on the set *V*
_{λ}. In the proof we give a new description of the orthogonal complement (*S*
^{λ})^{⊥} (with respect to the James scalar product) of the Specht module *S*
^{λ}. As applications, a basis of (*S*
^{λ})^{⊥} is exhibited, and we obtain a combinatorial description of the Hilbert function of *V*
_{λ..} Our approach gives also the deglex standard monomials of *V*
_{λ}, and hence provides a new proof of a result of A. M. Garsia and C. Procesi [10].

Let
*n*
be an arbitrary integer, let
*p*
be a prime factor of
*n*
. Denote by
*ω*
_{1}
the
*p*
^{th}
primitive unity root,

*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.