A Note on Non-jumping Numbers for <italic>r</italic>-Uniform Hypergraphs.

A real number α∈[0,1)<inline-graphic></inline-graphic> is a jump for an integer r≥2<inline-graphic></inline-graphic> if there exists a constant c>0<inline-graphic></inline-graphic> such that any number in (α,α+c]<inline-graphic></inline-graphic> cannot be the Turán density of a family of <italic>r</italic>-uniform graphs. Erdős and Stone showed that every number in [0,1) is a jump for r=2<inline-graphic></inline-graphic>. Erdős asked whether the same is true for r≥3<inline-graphic></inline-graphic>. Frankl and Rödl gave a negative answer by showing the existence of non-jumps for r≥3<inline-graphic></inline-graphic>. Recently, Baber and Talbot showed that every number in [0.2299,0.2316)⋃[0.2871,827)<inline-graphic></inline-graphic> is a jump for r=3<inline-graphic></inline-graphic> using Razborov’s flag algebra method. Pikhurko showed that the set of non-jumps for every r≥3<inline-graphic></inline-graphic> has cardinality of the continuum. But, there are still a lot of unknowns regarding jumps for hypergraphs. In this paper, we show that 1+r-1lr-1-rlr-2<inline-graphic></inline-graphic> is a non-jump for r≥4<inline-graphic></inline-graphic> and l≥3<inline-graphic></inline-graphic> which generalizes some earlier results. We do not know whether the same result holds for r=3<inline-graphic></inline-graphic>. In fact, when r=3<inline-graphic></inline-graphic> and l=3<inline-graphic></inline-graphic>, 1+r-1lr-1-rlr-2=29<inline-graphic></inline-graphic>, and determining whether 29<inline-graphic></inline-graphic> is a jump or not for r=3<inline-graphic></inline-graphic> is perhaps the most important unknown question regarding this subject. Erdős offered $500 for answering this question. [ABSTRACT FROM AUTHOR]