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

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 $r$.