Turán 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 $k\le r+1$ (it is easy to see that such a hypergraph is unique up to isomorphism). The known general bounds on its Turán density are $\pi (H_k^r)\le \frac{k-2}{r}$ for all $k\ge 3$, and $\pi (H_3^r)\ge 2^{1-r}$ for $k=3$. We prove that $\pi (H_k^r)\ge (C_k-o(1))r^{-(1+\frac{1}{k-2})}$ as $r\to \infty$. In the case $k=3$, we prove $\pi (H_3^r)\ge (1.7215-o(1))r^{-2}$ as $r\to \infty$, and $\pi (H_3^r)\ge r^{-2}$ for all $r$.