A note on generalized crowns in linear r-graphs

Abstract

An $r$-graph $H$ is a hypergraph consisting of a nonempty set of vertices $V$ and a collection of $r$-element subsets of $V$ we refer to as the edges of $H$. An $r$-graph $H$ is called linear if any two edges of $H$ intersect in at most one vertex. Let $F$ and $H$ be two linear $r$-graphs. If $H$ contains no copy of $F$, then $H$ is called $F$-free. The linear Turán number of $F$, denoted by $ex_r^{lin}(n,F)$, is the maximum number of edges in any $F$-free $n$-vertex linear $r$-graph. The crown $C_{13}$ (or $E_4$) is a linear 3-graph which is obtained from three pairwise disjoint edges by adding one edge that intersects all three of them in one vertex. In 2022, Gyárfás, Ruszinkó and Sárközy initiated the study of $ex_3^{lin}(n,F)$ for different choices of an acyclic 3-graph $F$. They determined the linear Turán numbers for all linear 3-graphs with at most 4 edges, except the crown. They established lower and upper bounds for $ex_3^{lin}(n,C_{13})$. In fact, their lower bound on $ex_3^{lin}(n,C_{13})$ is essentially tight, as was shown in a recent paper by Tang, Wu, Zhang and Zheng. In this paper, we generalize the notion of a crown to linear $r$-graphs for $r\ge 3$, and also generalize the above results to linear $r$-graphs.