Heavy Transversals and Indecomposable Hypergraphs

A weighted hypergraph is a hypergraph H = (V, E) with a weighting function $w:V \to R$, where R is the set of reals. A multiset S ⊆ V generates a partial hypergraph H ^S with edges ${\left\{ {e \in E{\kern 1pt} :{\kern 1pt} {\left| {e \cap S} \right|} > w{\left( S \right)}} \right\}}$, where both the cardinality ${\left| {e \cap S} \right|}$ and the total weight w(S) are counted with multiplicities of vertices in S. The transversal number of H is represented by τ(H). We prove the following: there exists a function f(n) such that, for any weighted n-Helly hypergraph H, τ(H ^B) ≤ 1, for all multisets B ⊆ V if and only if τ(H ^A) ≤ 1, for all multisets A ⊆ V with ${\left| A \right|} \leqslant f{\left( n \right)}$. We provide lower and upper bounds for f(n) using a link between indecomposable hypergraphs and critical weighted n-Helly hypergraphs.