Improved Upper Bound on the Linear Tur\'an Number of the Crown

Abstract

A linear $3$-graph is a set of vertices along with a set of edges, which are three element subsets of the vertices, such that any two edges intersect in at most one vertex. The crown, $C$, is a specific $3$-graph consisting of three pairwise disjoint edges, called jewels, along with a fourth edge intersecting all three jewels. For a linear $3$-graph, $F$, the linear Tur\'an number, $ex(n,F)$, is the maximum number of edges in any linear $3$-graph that does not contain $F$ as a subgraph. Currently, the best known bounds on the linear Tur\'an number of the crown are \[ 6 \Big \lfloor \frac{n-3}{4}\Big \rfloor \leq ex(n, C) \leq 2n. \] In this paper, the upper bound is improved to $ex(n,C) < \frac{5n}{3}$.