The Finite Projective Plane and the 5-Uniform Linear Intersecting Hypergraphs with Domination Number Four

The matching number alpha'(H) of a hypergraph H is the size of a largest matching in H, where a matching is a set of pairwise disjoint edges in H. A dominating set in H is a subset D of vertices of H such that for every v is an element of V(H) \ D there exists u is an element of D such that u and v lie in an edge of H, and the domination number of H, denoted by gamma(H), is the minimum cardinality of a dominating set in H. It was shown that a Ryser-like inequality gamma(H) <= (r - 1) alpha' (H) holds for hypergraphs H of rank r. In particular, for intersecting hypergraphs H of rank r, gamma(H) <= r - 1, since alpha'(H) = 1. The linear intersecting hypergraphs of rank 2 <= r <= 4 achieving the equality gamma(H) = r - 1 have been characterized. In this paper we show that all the 5- uniform linear intersecting hypergraphs H with equality gamma(H) = r - 1 are generated by the finite projective plane of order three.