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

• Published in: Discrete mathematics Vol. 345; no. 11; p. 113025
• Main Authors: Gyárfás, András, Sárközy, Gábor N.
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.11.2022
• Subjects: Linear triple systems; Ramsey numbers; Steiner triple systems; Turan numbers; Ramsey numbers; Turan numbers; Linear triple systems; Steiner triple systems
• ISSN: 0012-365X; 1872-681X
• DOI: 10.1016/j.disc.2022.113025

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) Turán 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 3×3 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)n26 with some c>0, separating it from the conjectured asymptotic of ex(n,G3×3), where G3×3, the grid, formed by three rows and three columns of a 3×3 point matrix.