A Note on Non-jumping Numbers for r-Uniform Hypergraphs

摘要

A real number is a jump for an integer if there exists a constant such that any number in cannot be the Turan density of a family of r-uniform graphs. ErdAs and Stone showed that every number in [0,1) is a jump for . ErdAs asked whether the same is true for . Frankl and Rodl gave a negative answer by showing the existence of non-jumps for . Recently, Baber and Talbot showed that every number in is a jump for using Razborov's flag algebra method. Pikhurko showed that the set of non-jumps for every has cardinality of the continuum. But, there are still a lot of unknowns regarding jumps for hypergraphs. In this paper, we show that is a non-jump for and which generalizes some earlier results. We do not know whether the same result holds for . In fact, when and , , and determining whether is a jump or not for is perhaps the most important unknown question regarding this subject. ErdAs offered $500 for answering this question.