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 everyBX. We say that α defines a 0-configuration with respect toA⫅2x 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<kn and letA={AX: |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).

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Deza, M., Frankl, P. On the vector space of 0-configurations. Combinatorica 2, 341–345 (1982). https://doi.org/10.1007/BF02579430

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AMS subject classification (1980)

  • 05 C 65
  • 05 C 35
  • 15 A 03