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arXiv · 2109.10520

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

Chaoliang Tang·Hehui Wu·Shengtong Zhang·Zeyu Zheng
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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.

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Chaoliang Tang, Hehui Wu, Shengtong Zhang, Zeyu Zheng. 2021-09-22. Note on the Tur\'an number of the $3$-linear hypergraph $C_{13}$. https://arxiv.org/abs/2109.10520

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