Non-jumping Turán densities of hypergraphs.

A real number α ∈ [ 0 , 1) is a jump for an integer r ≥ 2 if there exists c > 0 such that no number in (α , α + c) can be the Turán density of a family of r -uniform graphs. A classical result of Erdős and Stone [8] implies that every number in [ 0 , 1) is a jump for r = 2. Erdős [6] also showed that every number in [ 0 , r ! / r r) is a jump for r ≥ 3 and asked whether every number in [ 0 , 1) is a jump for r ≥ 3. Frankl and Rödl [9] gave a negative answer by showing a sequence of non-jumps for every r ≥ 3. After this, Erdős modified the question to be whether r ! r r is a jump for r ≥ 3 ? What's the smallest non-jump? Frankl, Peng, Rödl and Talbot [10] showed that 5 r ! 2 r r is a non-jump for r ≥ 3. However, Baber and Talbot [1] showed that there are more jumps by proving that every α ∈ [ 0.2299 , 0.2316) ∪ [ 0.2871 , 8 27) is a jump for r = 3. However, whether r ! r r is a jump for r ≥ 3 remains open, and 5 r ! 2 r r has remained the known smallest non-jump for r ≥ 3. In this paper, we give a smaller non-jump by showing that 54 r ! 25 r r is a non-jump for r ≥ 3. Furthermore, we give infinitely many irrational non-jumps for every r ≥ 3. [ABSTRACT FROM AUTHOR]