Jumping densities for \(3\)-hypergraphs

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 \( r=2\).

A problem on jumps in hypergraphs ($500) (proposed by Erdös[1])

Prove or disprove that any 3-uniform hypergraph with \(n > n_0(\epsilon)\) vertices and at least \((1/27 + \epsilon) n^3\) edges contains a subgraph on \(m\) vertices and at least \((1/27 + c) m^3\) edges where \(c >0\) does not depend on \(\epsilon \) and \(m\).

Originally, Erdös[1] asked the question of determining such a jump for the maximum density of subgraphs in hypergraphs with any given edge density. However, Frankl and Rödl [2] gave an example showing for hypergraphs with a certain edge density, there is no such jump for the density of subgraphs. Still, the original question for \( 3\)-uniform hypergraphs as described above remains open.

Bibliography
1 P. Erdös, Problems and results in graph theory and combinatorial analysis, Graph theory and related topics (Proc. Conf., Univ. Waterloo, Waterloo, Ont., 1977), 153-163, Academic Press, New York-London, 1979.

2 P. Frankl and V. Rödl, Hypergraphs do not jump, Combinatorica 4 (1984), 149-159.