In a survey by Füredi and Simonovits called "The history of degenerate (bipartite) extremal graph problems," Theorem 10.5 states the following:

Let $\mathcal K = K^{(r)}(a_1, \dots, a_r)$ be the complete $r$-partite $r$-graph with vertex classes of sizes $a_1, \dots, a_r$, with $|a_1| = t$. There exists a value $m = O(n^{r-{1}/{t^{r-1}}})$ such that every $r$-graph on $n$ vertices with $m$ edges contains a copy of $\mathcal K$.

It is not clear from the notation whether or not the host graph must also be $r$-partite, but for asymptotic results, this typically does not matter. My main concern is that Füredi and Simonovits cite the paper "On some extremal problems in graph theory" by Erdős (1965, Israel J. Math.), but the paper that they cite doesn't actually contain anything about hypergraphs. There is another paper by Erdős called "On extremal problems of graphs and generalized graphs" (1964, Israel J. Math) which deals with this same problem, but in the special case that $a_1 = \dots = a_r$. However, directly applying the method from this second paper to the general problem seems to give $r - \frac{1}{\prod_{i = 1}^{r-1} a_i}$ in the exponent of $n$. This brings me to my questions.

- Is this Theorem 10.5 even true?
- If this theorem is true, what is the correct reference for the result?

I'd be grateful for anyone's help. Thank you.