2021/12/30 by Yan, Zilong, Peng, Yuejian
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2112.14943
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 Erd\H os and Stone \citeES implies that that every number in [0, 1) is a jump for r=2. Erd\H os \citeE64 also showed that every number in [0, r!/rr) is a jump for r≥ 3 and asked whether every number in [0, 1) is a jump for r≥ 3. Frankl and Rödl \citeFR84 gave a negative answer by showing a sequence of non-jumps for every r≥ 3. After this, Erd\H os modified the question to be whether (r!)/(rr) is a jump for r≥ 3? What's the smallest non-jump? Frankl, Peng, Rödl and Talbot \citeFPRT showed that 5r!\over 2rr is a non-jump for r≥ 3. Baber and Talbot \citeBT0 showed that every α∈[0.2299, 0.2316)∪ [0.2871, (8)/(27)) is a jump for r=3. Pikhurko \citePikhurko2 showed that the set of all possible Turán densities of r-uniform graphs has cardinality of the continuum for r≥ 3. However, whether (r!)/(rr) is a jump for r≥ 3 remains open, and (5r!)/(2rr) has remained the known smallest non-jump for r≥ 3. In this paper, we give a smaller non-jump by showing that 54r!\over 25rr is a non-jump for r≥ 3. Furthermore, we give infinitely many irrational non-jumps for every r≥ 3.