Geodesic
On the Turán number of the linear \(3\)-graph \(C_{13}\)
Chaoliang Tang1 ; Hehui Wu2 ; Shengtong Zhang3 ; Zeyu Zheng1
1School of Mathematical Sciences, Fudan University
2Shanghai Center for Mathematical Sciences, Fudan University
3Massachusetts Institute of Technology
The electronic journal of combinatorics, Tome 29 (2022) no. 3
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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 $$\lvert E(G) \rvert \leq \frac{3(n - s)}{2}$$ 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 Turán number for linear $3$-graphs with at most $4$ edges.
DOI : 10.37236/10775
Classification : 05C30, 05C35, 05C65, 05D05, 05B07
Mots-clés : linear Turán number for linear 3-graphs, Steiner triple systems, Erdős-Sós conjecture

Chaoliang Tang  1   ; Hehui Wu  2   ; Shengtong Zhang  3   ; Zeyu Zheng  1

1 School of Mathematical Sciences, Fudan University
2 Shanghai Center for Mathematical Sciences, Fudan University
3 Massachusetts Institute of Technology 
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     author = {Chaoliang  Tang and Hehui  Wu and Shengtong Zhang and Zeyu Zheng},
     title = {On the {Tur\'an} number of the linear \(3\)-graph {\(C_{13}\)}},
     journal = {The electronic journal of combinatorics},
     year = {2022},
     volume = {29},
     number = {3},
     doi = {10.37236/10775},
     zbl = {1496.05081},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/10775/}
}
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Chaoliang  Tang; Hehui  Wu; Shengtong Zhang; Zeyu Zheng. On the Turán number of the linear \(3\)-graph \(C_{13}\). The electronic journal of combinatorics, Tome 29 (2022) no. 3. doi: 10.37236/10775

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