On the vector space of 0-configurations

Abstract

Let α be a rational-valued set-function on then-element sexX i.e. α(B) εQ for everyB ⫅X. We say that α defines a 0-configuration with respect toA⫅2^x if for everyA εA we have$\mathop \Sigma \limits_{A \subseteqq B \subseteqq X}$ α(B)=0. The 0-configurations form a vector space of dimension 2ⁿ − |A| (Theorem 1). Let 0 ≦t<k ≦n and letA={A ⫅X: |A| ≦t}. We show that in this case the 0-configurations satisfying α(B)=0 for |B|>k form a vector space of dimension$\mathop \Sigma \limits_{t< i \leqq k} \left( {\begin{array}{*{20}c} n \\ i \\ \end{array} } \right)$, we exhibit a basis for this space (Theorem 4). Also a result of Frankl, Wilson [3] is strengthened (Theorem 6).