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

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Yuejian Peng

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


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.


[1] D.P. Bertsekas, Constrained Optimization and Lagrange Multiplier Methods (Academic Press, New York, NY, 1982).
[2] P. Erdös, On extremal problems of graphs and generalized graphs, Israel J. Math. 2 (1964) 183-190, doi: 10.1007/BF02759942.
[3] P. Erdös and M. Simonovits, A limit theorem in graph theory, Studia Sci. Mat. Hung. Acad. 1 (1966) 51-57.
[4] P. Erdös and A.H. Stone, On the structure of linear graphs, Bull. Amer. Math. Soc. 52 (1946) 1087-1091, doi: 10.1090/S0002-9904-1946-08715-7.
[5] P. Frankl and Z. Füredi, Extremal problems whose solutions are the blow-ups of the small Witt-designs, J. Combin. Theory (A) 52 (1989) 129-147, doi: 10.1016/0097-3165(89)90067-8.
[6] P. Frankl and V. Rödl, Hypergraphs do not jump, Combinatorica 4 (1984) 149-159, doi: 10.1007/BF02579215.
[7] P. Frankl, Y. Peng, V. Rödl and J. Talbot, A note on the jumping constant conjecture of Erdös, J. Combin. Theory (B) 97 (2007) 204-216, doi: 10.1016/j.jctb.2006.05.004.
[8] G. Katona, T. Nemetz and M. Simonovits, On a graph problem of Turán, Mat. Lapok 15 (1964) 228-238.
[9] T.S. Motzkin and E.G. Straus, Maxima for graphs and a new proof of a theorem of Turán, Canad. J. Math. 17 (1965) 533-540, doi: 10.4153/CJM-1965-053-6.
[10] Y. Peng, Non-jumping numbers for 4-uniform hypergraphs, Graphs and Combinatorics 23 (2007) 97-110, doi: 10.1007/s00373-006-0689-5.
[11] Y. Peng, Using Lagrangians of hypergraphs to find non-jumping numbers (I), submitted.
[12] Y. Peng, Using Lagrangians of hypergraphs to find non-jumping numbers (II), Discrete Math. 307 (2007) 1754-1766, doi: 10.1016/j.disc.2006.09.024.
[13] J. Talbot, Lagrangians of hypergraphs, Combinatorics, Probability & Computing 11 (2002) 199-216, doi: 10.1017/S0963548301005053.

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