Non-jumps of hypergraphs

A density $\alpha \in [0,1)$ is a jump for $r$ if there is some $c>0$ such that there does not exist a family of $r$-uniform hypergraphs $\mathcal{F}$ with Turán density $\pi (\mathcal{F})$ in $(\alpha ,\alpha +c)$. Erdös conjectured that all $\alpha \in [0,1)$ are jumps for any $r$. This was disproven by Frankl and Rödl when they provided examples of non-jumps. In this paper, we provide a method for finding non-jumps for $r=3$ using patterns. As a direct consequence, we find a few more examples of non-jumps for $r=3$.