Turán densities of hypergraph augmentations

Abstract

For a $t$-graph $F$ and an integer $r\ge t$, the \emph{$r$-augmentation} $F(r)$ is the $r$-graph obtained by adding the same set of $r-t$ new vertices to every edge of $F$. We investigate the behaviour of the \emph{Turán density $π(F(r))$}, which is the asymptotically maximum edge density of a large $F(r)$-free $r$-graph, as a function of $r\to\infty$. Let $H_k^r$ be the $r$-augmentation of the complete $(k-1)$-graph on $k$ vertices; equivalently, $H_k^r$ is the (unique up to isomorphism) $r$-graph with $r+1$ vertices and $k$ edges. This paper determines the Turán density of each $r$-graph $H_k^r$ within a constant factor and estimates the lower bounds on $π(H_k^r)$ coming from the circular construction of Sidorenko. In particular, it is shown that if $k/\log r\to\infty$ then $π(H_k^r)=(1+o(1))k/r$. Also, we consider augmentations of graphs (that is, the case $t=2$) and prove that $π(F(r))\ge(\log r)^{-1-o(1)}/r$ for every 3-chromatic graph $F$, which comes close to the known upper bound $1/r$. Note that, for all other graphs $F$, the function $π(F(r))$ is known to be either identically 0 or uniformly bounded away from 0.