Graphs and Combinatorics

, Volume 34, Issue 3, pp 489–499 | Cite as

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

Original Paper


A real number \(\alpha \in [0,1)\) is a jump for an integer \(r\ge 2\) if there exists a constant \(c>0\) such that any number in \((\alpha , \alpha +c]\) cannot be the Turán density of a family of r-uniform graphs. Erdős and Stone showed that every number in [0,1) is a jump for \(r=2\). Erdős asked whether the same is true for \(r\ge 3\). Frankl and Rödl gave a negative answer by showing the existence of non-jumps for \(r\ge 3\). Recently, Baber and Talbot showed that every number in \([0.2299,0.2316)\bigcup [0.2871, \frac{8}{27})\) is a jump for \(r=3\) using Razborov’s flag algebra method. Pikhurko showed that the set of non-jumps for every \(r\ge 3\) has cardinality of the continuum. But, there are still a lot of unknowns regarding jumps for hypergraphs. In this paper, we show that \(1+\frac{r-1}{l^{r-1}}-\frac{r}{l^{r-2}}\) is a non-jump for \(r\ge 4\) and \(l\ge 3\) which generalizes some earlier results. We do not know whether the same result holds for \(r=3\). In fact, when \(r=3\) and \(l=3\), \(1+\frac{r-1}{l^{r-1}}-\frac{r}{l^{r-2}}={2 \over 9}\), and determining whether \({2 \over 9}\) is a jump or not for \(r=3\) is perhaps the most important unknown question regarding this subject. Erdős offered $500 for answering this question.


Extremal problems in hypergraphs Turán density Erdős jumping constant conjecture Lagrangians of uniform hypergraphs 


  1. 1.
    Baber, R., Talbot, J.: Hypergraphs do jump. Combin. Probab. Comput. 20(2), 161–171 (2011)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Erdős, P.: On extremal problems of graphs and generalized graphs. Isr. J. Math. 2, 183–190 (1964)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Erdős, P., Simonovits, M.: A limit theorem in graph theory. Studia Sci. Mat. Hungar. Acad. 1, 51–57 (1966)MathSciNetMATHGoogle Scholar
  4. 4.
    Erdős, P., Stone, A.H.: On the structure of linear graphs. Bull. Am. Math. Soc. 52, 1087–1091 (1946)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Frankl, P., Peng, Y., Rödl, V., Talbot, J.: A note on the jumping constant conjecture of Erdös. J. Combin. Theory Ser. B. 97, 204–216 (2007)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Frankl, P., Rödl, V.: Hypergraphs do not jump. Combinatorica 4, 149–159 (1984)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Gu, R., Li, X., Qin, Z., Shi, Y., Yang, K.: Non-jumping numbers for 5-uniform hypergraphs. Appl. Math. Comput. 317, 234–251 (2018)MathSciNetGoogle Scholar
  8. 8.
    Peng, Y.: Non-jumping numbers for 4-uniform hypergraphs. Graphs Combin. 23(1), 97–110 (2007)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Peng, Y.: Using lagrangians of hypergraphs to find non-jumping numbers I. Ann. Combin. 12, 307–324 (2008)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Peng, Y.: Using Lagrangians of hypergraphs to find non-jumping numbers (II). Discrete Math. 307, 1754–1766 (2007)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Peng, Y.: On substructure densities of hypergraphs. Graphs Combin. 25(4), 583–600 (2009)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Peng, Y.: On jumping densities of hypergraphs. Graphs Combin. 25, 759–766 (2009)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Pikhurko, O.: On possible turán densities. Isr. J. Math. 201, 415–454 (2014)CrossRefMATHGoogle Scholar

Copyright information

© Springer Japan KK, part of Springer Nature 2018

Authors and Affiliations

  1. 1.College of Mathematics and EconometricsHunan UniversityChangshaPeople’s Republic of China
  2. 2.Institute of MathematicsHunan UniversityChangshaPeople’s Republic of China

Personalised recommendations