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

Shengtong Zhang

Published 2021-09-22Version 1

Let $C_{13}$ be the $3$-linear hypergraph 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 $$\text{ex}(n, C_{13}) \leq \frac{3n}{2}.$$ This essentially completes the determination of linear Tur\'an number for $3$-linear hypergraphs with at most $4$ edges.