  In this paper, we consider the Tur\'an problems on $\{1,3\}$-hypergraphs. We
prove that a $\{1, 3\}$-hypergraph is degenerate if and only if it's $H^{\{1,
3\}}_5$-colorable, where $H^{\{1, 3\}}_5$ is a hypergraph with vertex set
$V=[5]$ and edge set $E=\{\{2\}, \{3\}, \{1, 2, 4\}, \{1, 3, 5\}, \{1, 4,
5\}\}.$ Using this result, we further prove that for any finite set $R$ of
distinct positive integers, except the case $R=\{1, 2\}$, there always exist
non-trivial degenerate $R$-graphs. We also compute the Tur\'an densities of
some small $\{1,3\}$-hypergraphs.
