Note on the Turán number of the $3$-linear hypergraph $C_{13}$

Abstract: Let the crown $C_{13}$ be the linear $3$-graph on $9$ vertices ${a,b,c,d,e,f,g,h,i}$ with edges $$E = {{a,b,c}, {a, d,e}, {b, f, g}, {c, h,i}}.$$ Proving a conjecture of Gy\'arf\'as et. al., we show that for any crown-free linear $3$-graph $G$ on $n$ vertices, its number of edges satisfy $$\lvert E(G) \rvert \leq \frac{3(n - s)}{2}$$ where $s$ is the number of vertices in $G$ with degree at least $6$. This result, combined with previous work, essentially completes the determination of linear Tur\'an number for linear $3$-graphs with at most $4$ edges.