Turán number of generalized triangles

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• 作者：S. Norin ; L. Yepremyan ^{yepremyan@maths.ox.ac.uk" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Generalized triangle ; Blowups ; Turá ; n number ; Steiner systems ; Symmetrization ; Stability ; Lagrangian function ; Weighted hypergraphs
• 刊名：Journal of Combinatorial Theory, Series A
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：146
• 期：Complete
• 页码：312-343
• 全文大小：558 K

文摘

The family Σ_r${\mathrm{\Sigma }}_r$ consists of all r  -graphs with three edges D₁,D₂,D₃$D_1,D_2,D_3$ such that |D₁∩D₂|=r−1$|D_1\cap D_2|=r-1$ and D₁△D₂⊆D₃$D_1△D_2\subseteq D_3$. A generalized triangle  , T_r∈Σ_r${\mathcal{T}}_r\in {\mathrm{\Sigma }}_r$ is an r  -graph on {1,2,…,2r−1}$\{1,2,\dots ,2r-1\}$ with three edges D₁,D₂,D₃$D_1,D_2,D_3$, such that D₁={1,2,…,r−1,r},D₂={1,2,…,r−1,r+1}$D_1=\{1,2,\dots ,r-1,r\},D_2=\{1,2,\dots ,r-1,r+1\}$ and D₃={r,r+1,…,2r−1}$D_3=\{r,r+1,\dots ,2r-1\}$.

Frankl and Füredi conjectured that for all r≥4$r\ge 4$, ex(n,Σ_r)=ex(n,T_r)$\mathrm{ex}(n,{\mathrm{\Sigma }}_r)=\mathrm{ex}(n,{\mathcal{T}}_r)$ for all sufficiently large n   and they also proved it for r=3$r=3$. Later, Pikhurko showed that the conjecture holds for r=4$r=4$. In this paper we determine ex(n,T₅)$\mathrm{ex}(n,{\mathcal{T}}_5)$ and ex(n,T₆)$\mathrm{ex}(n,{\mathcal{T}}_6)$ for sufficiently large n  , proving the conjecture for r=5,6$r=5,6$.