The linear Turan number of small triple systems or why is the wicket interesting?

A linear triple system is a 3-uniform hypergraph H = (V, E), where E is a set of three element subsets of V such that any two edges intersect in at most one vertex. For linear triple systems H, F we say that H is F-free if H does not contain any subsystem isomorphic to F. We consider F fixed and call it a configuration. The (linear) Turan number exL(n, F) (or simply just ex(n, F)) of a configuration F is the maximum number of edges in F-free linear triple systems with n vertices. Here we call attention to some properties of the wicket W, formed by three rows and two columns of a 3 x 3 point matrix. On one hand we show that the problem whether ex(n, F) = o(n2) can be decided for all configurations with at most five edges, except for F = W, which remains undecided. On the other hand we prove that ex(n, W ) <= (1-c)n(2)/6 with some c > 0, separating it from the conjectured asymptotic of ex(n, G3x3), where G3x3, the grid, formed by three rows and three columns of a 3 x 3 point matrix. (c) 2022 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).