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 Gyárfás et. al., we show that for any crown-free linear $3$-graph $G$ on $n$ vertices, its number of edges satisfy $$|E(G)|\le \frac{3(n-s)}{2}$$ 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 Turán number for linear $3$-graphs with at most $4$ edges.

5 pages, 1 figures. Correct Typos, and add acknowledgement to Professor Gyarfas

References in corpus (1)

• Crowns in linear $3$-graphs