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 σ_r(m,f) =. lim sup_n →∞|{e : (n,e) →_r (m,f)}|/nr. . 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, mr). 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.