On the Turán density of \(\{1, 3\}\)-hypergraphs (Q1732027)

Summary: 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.