Advertisement

Combinatorica

, Volume 4, Issue 2–3, pp 149–159 | Cite as

Hypergraphs do not jump

  • Peter Frankl
  • Vojtěch Rödl
Article

Abstract

The number α, 0≦α≦1, is a jump forr if for any positive ε and any integerm,mr, anyr-uniform hypergraph withn>n o (ε,m) vertices and at least (α+ε)\(\left( {\begin{array}{*{20}c} n \\ r \\ \end{array} } \right)\) edges contains a subgraph withm vertices and at least (α+c)\(\left( {\begin{array}{*{20}c} m \\ r \\ \end{array} } \right)\) edges, wherec=c(α) does not depend on ε andm. It follows from a theorem of Erdös, Stone and Simonovits that forr=2 every α is a jump. Erdös asked whether the same is true forr≧3. He offered $ 1000 for answering this question. In this paper we give a negative answer by showing that\(1 - \frac{1}{{l^{r - 1} }}\) is not a jump ifr≧3,l>2r.

AMS subject classification (1980)

05 C 65 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    P. Erdös, On extremal problems of graphs and generalized graphs,Israel J. Math. 2 (1965), 183–190.Google Scholar
  2. [2]
    P. Erdös, Problems and Results on Graph and Hypergraphs, Similarities and Differences.Google Scholar
  3. [3]
    P. Erdös andM. Simonovits, A limit theorem in graph theory,Studia Sci. Mat. Hung. Acad. 1 (1966), 51–57.zbMATHGoogle Scholar
  4. [4]
    P. Erdös andA. H. Stone, On the structure of linear graphs,Bull. Amer. Math. Soc. 52 (1946), 1087–1091.zbMATHMathSciNetCrossRefGoogle Scholar
  5. [5]
    V. Jarník,Differentialrechnung, Prague 1956.Google Scholar
  6. [6]
    G. Katona, T. Nemetz andM. Simonovits, On a graph-problem of Turán,Mat. Lapok 15 (1964), 228–238.zbMATHMathSciNetGoogle Scholar
  7. [7]
    P. Turán, On an extremal problem in graph theory (in Hungarian),Mat. Fiz. Lapok 48 (1941), 436–452.zbMATHMathSciNetGoogle Scholar

Copyright information

© Akadémiai Kiadó 1984

Authors and Affiliations

  • Peter Frankl
    • 1
  • Vojtěch Rödl
    • 2
  1. 1.CNRSParisFrance
  2. 2.Dept. Math.FJFI ČVUTPraha 1ČSSR

Personalised recommendations