On the thresholds of degenerate hypergraphs

An $n$-vertex $k$-uniform hypergraph $G$ is $(d,\alpha)$-degenerate if $m_1(G)\le{d}$ and there exists a constant $\varepsilon >0$ such that for every subset $U\subseteq{V(G)}$ with size $2\le|U|\le{\varepsilon n}$, we have $e\left(G[U]\right)\le{d\left(|U|-1\right)-\alpha}$. These hypergraphs include many natural graph classes, such as the degenerate hypergraphs, the planar graphs, and the power of cycles. In this paper, we consider the threshold of the emergence of a $(d,\alpha)$-degenerate hypergraph with bounded maximum degree in the Erd\H{o}s-R\'enyi model. We show that its threshold is at most $n^{-1/d}$, improving previous results of Riordan and Kelly-M\"uyesser-Pokrovskiy.