Exact Tur\'{a}n densities in triple systems

TL;DR

This paper establishes exact Turán densities for specific 3-graph configurations, confirming conjectures and addressing open problems in extremal combinatorics related to triple systems.

Contribution

It proves new exact Turán density results for certain 3-graphs, including confirming a conjecture and solving open problems in the field.

Findings

Confirmed conjecture of Shi on Turán densities.

Provided exact densities for specific triple systems.

Addressed open problems posed by Mubayi and R"odl.

Abstract

In this paper, we prove several new Tur\'{a}n density results for $3$-graphs. We show: $\pi(C_4^3, \mathrm{complement\ of\ } F_5) = 2\sqrt{3} - 3$, $\pi(F_{3,2}, C_5^{3-}) = \frac{2}{9}$, and $\pi(F_{3,2}, \mathrm{induced\ complement\ of\ } F_{3,2}) = \frac{3}{8}$. The first result confirms the conjecture of Shi~[On Tur\'an denisties of small triple graphs, European J. Combin. 52 (2016) 95-102]. The other results give several special non-principal family posed by Mubayi and R\"odl~[On the Tur\'an number of triple systems, J. Combin. Theory A. 100 (2002) 135-152].