Turán Number of Generalized Triangles

Sergey Norin; Liana Yepremyan

paper

Turán Number of Generalized Triangles

arXiv:1501.01913

Abstract

The family $Σ_{r}$ consists of all $r$ -graphs with three edges $D_{1}, D_{2}, D_{3}$ such that $∣ D_{1} \cap D_{2} ∣ = r - 1$ and $D_{1} △ D_{2} \subseteq D_{3}$ . A generalized triangle, $T_{r} \in Σ_{r}$ is an $r$ -graph on ${1, 2, \dots, 2 r - 1}$ with three edges $D_{1}, D_{2}, D_{3}$ , such that $D_{1} = {1, 2, \dots, r - 1, r}, D_{2} = {1, 2, \dots, r - 1, r + 1}$ and $D_{3} = {r, r + 1, \dots, 2 r - 1} .$ Frankl and Füredi conjectured that for all $r \geq 4$ , $e x (n, Σ_{r}) = e x (n, T_{r})$ for all sufficiently large $n$ and they also proved it for $r = 3$ . Later, Pikhurko showed that the conjecture holds for $r = 4$ . In this paper we determine $e x (n, T_{5})$ and $e x (n, T_{6})$ for sufficiently large $n$ , proving the conjecture for $r = 5, 6$ .

arXiv comment31 pages