Tur\'an Number of Generalized Triangles

TL;DR

This paper determines the Turán numbers for generalized triangles $\\mathcal{T}_r$ for $r=5,6$ and large $n$, confirming a conjecture relating these to the extremal numbers of certain hypergraph families.

Contribution

It explicitly computes the Turán numbers for $\\mathcal{T}_5$ and $\\mathcal{T}_6$, proving the conjecture for these cases and advancing understanding of extremal hypergraph configurations.

Findings

Determined $ex(n,\mathcal{T}_5)$ for large $n$.

Determined $ex(n,\mathcal{T}_6)$ for large $n$.

Confirmed the conjecture for $r=5,6$.

Abstract

The family $\Sigma_r$ consists of all $r$-graphs with three edges $D_1,D_2,D_3$ such that $|D_1\cap D_2|=r-1$ and $D_1 \triangle D_2 \subseteq D_3$. A generalized triangle, $\mathcal{T}_r \in \Sigma_r$ is an $r$-graph on $\{1,2,\ldots,2r-1\}$ 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 $D_3 = \{r, r+1, \dots, 2r-1\}.$ Frankl and F\"{u}redi conjectured that for all $r\geq 4$, $ex(n,\Sigma_r) = ex(n,\mathcal{T}_r )$ 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 $ex(n,\mathcal{T}_5)$ and $ex(n,\mathcal{T}_6)$ for sufficiently large $n$, proving the conjecture for $r=5,6$.