Linear Turán numbers of acyclic triple systems

The Turán number of hypergraphs has been studied extensively. Here we deal with a recent direction, the linear Turán number, and restrict ourselves to linear triple systems, a collection of triples on a set of points in which any two triples intersect in at most one point. For a fixed linear triple system F, the linear Turán number exL(n,F) is the maximum number of triples in a linear triple system with n points that does not contain F as a subsystem. We initiate the study of the linear Turán number for an acyclicF. In this case exL(n,F) is linear in n and we aim for good bounds. Since the case of trees is already difficult for graphs (Erdős–Sós conjecture), we concentrate on matchings, paths and small trees. In case of matchings, where Mk is the set of k pairwise disjoint triples, we prove that for fixed k and large enough n, exL(n,Mk)=f(n,k) where f(n,k) is the maximum number of triples that can meet k−1 points in a linear triple system on n points. This is an analogue of an old result of Erdős on hypergraph matchings. For the k-edge linear path Pk we show (extending some standard path increasing methods used for graphs) that exL(n,Pk)≤1.5kn which is probably far from best possible. Finding exL(n,F) relates to difficult problems on Steiner triple systems and interesting even for small trees. For example, for P4, the path with four triples, exL(n,P4)≤4n3 with equality only for disjoint union of affine planes of order 3. On the other hand, for E4, the tree having three pairwise disjoint triples and a fourth one meeting all of them, we have bounds only: 6⌊n−34⌋≤exL(n,E4)≤2n.