Spreading linear triple systems and expander triple systems

Abstract

The existence of Steiner triple systems STS(n) of order n containing no nontrivial subsystem is well known for every admissible n. We generalize this result in two ways. First we define the expander property of 3-uniform hypergraphs and show the existence of Steiner triple systems which are almost perfect expanders.
Next we define the strong and weak spreading property of linear hypergraphs, and determine the minimum size of a linear triple system with these properties, up to a small constant factor. A linear triple system on a vertex set $V$ has the spreading, or respectively weakly spreading property if any sufficiently large subset V'\subset V contains a pair of vertices with which a vertex of V \ V' forms a triple of the system. Here the condition on $V'$ refers to |V'|>3 or V' is the support of more than one triples, respectively. This property is strongly connected to the connectivity of the structure the so-called influence maximisation.We also discuss how the results are related to Erdős' conjecture on locally sparse STSs, subsquare-free Latin-squares and possible applications in finite geometry.