On the Turán Density of $\{1, 3\}$-Hypergraphs

Abstract

In this paper, we consider the Turán 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 Turán densities of some small $\{1,3\}$-hypergraphs.