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 $H_5^{\{1,3\}}$-colorable, where $H_5^{\{1,3\}}$ 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.

18 pages