Unavoidable order-size pairs in hypergraphs — positive forcing density

Erdős, Füredi, Rothschild and Sós initiated a study of classes of graphs that forbid every induced subgraph on a given number m of vertices and number f of edges. Extending their notation to r-graphs, we write (n, e) →_r (m, f) if every r-graph G on n vertices with e edges has an induced subgraph on m vertices and f edges. The forcing density of a pair (m, f) is (Formula presented). In the graph setting it is known that there are infinitely many pairs (m, f) with positive forcing density. Weber asked if there is a pair of positive forcing density for r ≥ 3 apart from the trivial ones (m, 0) and (m, (^m_r)). Answering her question, we show that (6, 10) is such a pair for r = 3 and conjecture that it is the unique such pair. Further, we find necessary conditions for a pair to have positive forcing density, supporting this conjecture.