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Discussiones Mathematicae Graph Theory 27(2) (2007) 281-297
DOI: https://doi.org/10.7151/dmgt.1361

SUBGRAPH DENSITIES IN HYPERGRAPHS

Yuejian Peng

Department of Mathematics and Computer Science
Indiana State University
Terre Haute, IN, 47809, USA
e-mail: mapeng@isugw.indstate.edu

Abstract

Let r ≥ 2 be an integer. A real number α ∈ [0,1) is a jump for r if for any ε > 0 and any integer m ≥ r, any r-uniform graph with n > n₀(ε, m) vertices and density at least α+ε contains a subgraph with m vertices and density at least α+c, where c = c(α) > 0 does not depend on ε and m. A result of Erdös, Stone and Simonovits implies that every α ∈ [0,1) is a jump for r = 2. Erdös asked whether the same is true for r ≥ 3. Frankl and Rödl gave a negative answer by showing an infinite sequence of non-jumps for every r ≥ 3. However, there are still a lot of open questions on determining whether or not a number is a jump for r ≥ 3. In this paper, we first find an infinite sequence of non-jumps for r = 4, then extend one of them to every r ≥ 4. Our approach is based on the techniques developed by Frankl and Rödl.

Keywords: Erdös jumping constant conjecture, Lagrangian, optimal vector.

2000 Mathematics Subject Classification: 05D05, 05C65.

References

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Received 5 April 2006
Revised 18 September 2006
Accepted 18 September 2006