Description
The existence of Steiner triple systems STS(n) of order n containing no nontrivial subsystem is well known for every admissible n. We generalise this result in two ways.
First 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 strong, respectively weak spreading property if any subset $V' \subset V$ contains a pair of vertices with which a vertex of V \ V' forms a triple of the system, where V' has at least 4 vertices, or is the support of more than one triple, respectively.
We prove that the minimum size of a strong spreading linear hypergraph is significantly smaller than the size of a STS(n), while a weakly spreading linear triple system can be even of linear size.
Next we define the expander property of 3-uniform hypergraphs and show the existence of Steiner triple systems which are almost perfect expanders.
We also discuss how the results are related to Erdős' conjecture on locally sparse STSs, subsquare-free Latin-squares, hypergraph connectivity and models of information transmission.