Lower bound for hypergraph Turán numbers

A hypergraph \( H\) consists of a vertex set \( V\) together with a family \( E\) of subsets of \( V\), which are called the edges of \( H\). A \( r\)-uniform hypergraph, or \( r\)-graph, for short, is a hypergraph whose edge set consists of \( r\)-subsets of \( V\). A graph is just a special case of an \( r\)-graph with \( r=2\).

For a finite family \( \mathcal F\) of \( r\)-graphs, one can define the Turán number \( t_r(n,{\mathcal F})\) to be the largest integer \( t\) such that there is an \( r\)-graph on \( n\) vertices with \( t\) edges which does not contain any \( F \in {\mathcal F}\) as a subgraph.

Suppose \( \mathcal F\) consists of a single \( r\)-partite \( r\)-graph \( H\) with vertex set \( \cup_{i=1}^r A_i\) and \( \vert A_i\vert=t_i\). Erdös [1], [2] proved that

\begin{equation} t_r(n, H) \leq c n^{k-(t_1+\ldots+t_r)/(t_1\ldots t_r)} \end{equation}

where \( c\) is a constant depending only on \( r\) and the \( t_i\)'s.

Problem

For an \(r\)-partite \(r\)-graph \(H\) with vertex set \(\cup_{i=1}^r A_i\) and \(|A_i|=t_i\), prove that \[ t_r(n, H) \geq c n ^{k-(t_1+\ldots+t_r)/(t_1\ldots t_r)} \] for some constant \(c\).

Bibliography
1 P. Erdös. On extremal problems of graphs and generalized graphs. Israel J. Math. 2 (1964), 183-190.

2 Z. Füredi, Turán type problems, Surveys in Combinatorics, 1991 (Guildford, 1991), London Math. Soc. Lecture Notes Series 166, 253-300, Cambridge Univ. Press, Cambridge, 1991.