  Detection of a planted dense subgraph in a random graph is a fundamental
statistical and computational problem that has been extensively studied in
recent years. We study a hypergraph version of the problem. Let $G^r(n,p)$
denote the $r$-uniform Erd\H{o}s-R\'enyi hypergraph model with $n$ vertices and
edge density $p$. We consider detecting the presence of a planted
$G^r(n^\gamma, n^{-\alpha})$ subhypergraph in a $G^r(n, n^{-\beta})$
hypergraph, where $0< \alpha < \beta < r-1$ and $0 < \gamma < 1$. Focusing on
tests that are degree-$n^{o(1)}$ polynomials of the entries of the adjacency
tensor, we determine the threshold between the easy and hard regimes for the
detection problem. More precisely, for $0 < \gamma < 1/2$, the threshold is
given by $\alpha = \beta \gamma$, and for $1/2 \le \gamma < 1$, the threshold
is given by $\alpha = \beta/2 + r(\gamma - 1/2)$.
  Our results are already new in the graph case $r=2$, as we consider the
subtle log-density regime where hardness based on average-case reductions is
not known. Our proof of low-degree hardness is based on a conditional variant
of the standard low-degree likelihood calculation.
