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## Discussiones Mathematicae Graph Theory

2007 | 27 | 2 | 281-297
Tytuł artykułu

### Subgraph densities in hypergraphs

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Słowa kluczowe
EN
Kategorie tematyczne
Wydawca
Czasopismo
Rocznik
Tom
Numer
Strony
281-297
Opis fizyczny
Daty
wydano
2007
otrzymano
2006-04-05
poprawiono
2006-09-18
zaakceptowano
2006-09-18
Twórcy
autor
• Department of Mathematics and Computer Science, Indiana State University, Terre Haute, IN, 47809, USA
Bibliografia
• [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.
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Bibliografia
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