On Tur\'{a}n numbers of the complete $4$-graphs

TL;DR

This paper investigates the Turán numbers for complete 4-graphs, establishing an upper bound for the constant $t_*(4)$ that describes the asymptotic minimum edge count in r-graphs with bounded independence number.

Contribution

The paper provides a new upper bound for the asymptotic constant $t_*(4)$ in the Turán number formula for complete 4-graphs, advancing understanding of extremal combinatorics.

Findings

Proves that $t_*(4) < 0.706335$.

Confirms known values for $t_*(2)$ and conjectured value for $t_*(3)$.

Contributes to the characterization of Turán numbers for higher uniformities.

Abstract

The Tur\'{a}n number $T(n,\alpha+1,r)$ is the minimum number of edges in an $n$-vertex $r$-graph whose independence number does not exceed $\alpha$. For each $r\geq 2$, there exists $t_*(r)$ such that $T(n,\alpha+1,r) = t_*(r) \: n^r \: \alpha^{1-r} \: (1+o(1))$ as $\alpha / r \to\infty$ and $n / \alpha \to\infty$. It is known that $t_*(2) = 1/2$, and the conjectured value of $t_*(3)$ is $2/3$. We prove that $t_*(4) < 0.706335\:$.