The following is a theorem in the linked paper here.
To add some context, the authors also state:
Throughout this paper...$t^k$, the average degree, will always be an increasing function of $n$, the number of vertices of the hypergraph, i.e., $t = t(n) \to \infty$ with $n \to \infty$
How does one interpret the limit in the above sentence? Currently I am interpreting it as: if you add more and more vertices to your hypergraph, the $k$-th root of the average degree should also increase, but that doesn't seem to make much sense. I suppose my main trouble here is: are there particular $(k+1)$-uniform hypergraphs to which Theorem 2.4 cannot be applied, owing to the aforementioned limit?