Nagy Zoltan Lorant: Expander triple systems and spreading triple systems

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 expander property of 3-uniform hypergraphs and show the existence of Steiner triple systems which are almost perfect expanders. This result relies on additive combinatorial techniques.

    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 respectively weakly 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 triples, respectively.
We prove that the minimum size of a spreading linear triple system is significantly smaller than the size of a $\STS(n)$, namely 0.11n^2<\xi_{sp}(n)<  (0.139+o(1))n^2$, while a weakly spreading linear triple system can be even of linear size, as n-3\leq \xi_{wsp}(n)< 8/3 n+O(1) holds for its minimum size.
As a motivation, we discuss how these results are related to Erdős' conjecture on locally sparse STSs, subsquare-free Latin squares, hypergraph connectivity and possible applications in finite geometry.

joint work with Zoltán L. Blázsik