Independence in hypergraphs

Abstract

Suppose an integral function γ(|A|)⩾q₁ defined on the subsets of edges of a hypergraph (X,u,γ) satisfies the following two conditions: 1) any set W ⫅u such that |γA|⩾γ(|A|) for any A⫅W is matroidally independent; 2) if W is an independent set, then there exists a unique partitionW=T₁+ T₂+...+T_v such that |γTⁱ|=γ(|Tⁱ|),iε1:v, and for any A⫅W, |γA|⩾γ(|A|) there exists a T_i such that A⫅T_i. The form of such a function is found, in terms of parameters of generalized connected components, hypercycles, and hypertrees.