Complexity aspects of generalized Helly hypergraphs

In [V.I. Voloshin, On the upper chromatic number of a hypergraph, Australas. J. Combin. 11 (1995) 25–45], Voloshin proposed the following generalization of the Helly property. Let p⩾1, q⩾0 and s⩾0. A hypergraph H is (p,q)-intersecting when every partial hypergraph H′⊆H formed by p or less hyperedges has intersection of cardinality at least q. A hypergraph H is (p,q,s)-Helly when every partial (p,q)-intersecting hypergraph H′⊆H has intersection of cardinality at least s. In this work, we study the complexity of determining whether H is (p,q,s)-Helly.