Non-jumping densities of 3-uniform 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 with Turán density 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 $r = 3$.