Tur\'{a}n numbers of $r$-graphs on $r+1$ vertices

TL;DR

This paper investigates the Turán numbers of specific r-uniform hypergraphs with r+1 vertices, establishing new asymptotic lower bounds on their Turán densities as r grows large.

Contribution

It provides novel asymptotic lower bounds for the Turán density of hypergraphs with r+1 vertices, improving understanding of extremal hypergraph configurations.

Findings

Established that (H_k^r) b7 (C_k - o(1)) r^{-(1+1/(k-2))} as ra0d7a0a0a0

Proved (H_3^r) b7 (1.7215 - o(1)) r^{-2} as ra0d7a0a0a0

Showed (H_3^r) b7 r^{-2} for all r

Abstract

Let $H_k^r$ denote an $r$-uniform hypergraph with $k$ edges and $r+1$ vertices, where $k \leq r+1$ (it is easy to see that such a hypergraph is unique up to isomorphism). The known general bounds on its Tur\'{a}n density are $\pi(H_k^r) \leq \frac{k-2}{r}$ for all $k \geq 3$, and $\pi(H_3^r) \geq 2^{1-r}$ for $k=3$. We prove that $\pi(H_k^r) \geq (C_k - o(1)) \, r^{-(1+\frac{1}{k-2})}$ as $r\to\infty$. In the case $k=3$, we prove $\pi(H_3^r) \geq (1.7215 - o(1)) \, r^{-2}$ as $r\to\infty$, and $\pi(H_3^r) \geq r^{-2}$ for all $r$.