On the Tur\'an density of $\{1,3\}$-Hypergraphs

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$H^{{1, 3}}_5$isahypergraphwithvertexset$V=[5]$andedgeset$E={{2}, {3}, {1, 2, 4}, {1, 3, 5}, {1, 4, 5}}.$Usingthisresult,wefurtherprovethatforanyfiniteset$R$ 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.