Note on the Tur\'an number of the $3$-linear hypergraph $C_{13}$

TL;DR

This paper proves a bound on the number of edges in crown-free linear 3-graphs, confirming a conjecture and advancing the understanding of Turán numbers for small linear hypergraphs.

Contribution

It proves a conjecture by Gyárfás et al. on the maximum edges in crown-free linear 3-graphs, completing the Turán number determination for certain small hypergraphs.

Findings

Established an upper bound on edges in crown-free linear 3-graphs.

Confirmed the conjecture of Gyárfás et al. regarding Turán numbers.

Contributed to the classification of linear 3-graphs with up to 4 edges.

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.