# Subgraph densities in hypergraphs

Discussiones Mathematicae Graph Theory (2007)

- Volume: 27, Issue: 2, page 281-297
- ISSN: 2083-5892

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topYuejian Peng. "Subgraph densities in hypergraphs." Discussiones Mathematicae Graph Theory 27.2 (2007): 281-297. <http://eudml.org/doc/270519>.

@article{YuejianPeng2007,

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.},

author = {Yuejian Peng},

journal = {Discussiones Mathematicae Graph Theory},

keywords = {Erdös jumping constant conjecture; Lagrangian; optimal vector},

language = {eng},

number = {2},

pages = {281-297},

title = {Subgraph densities in hypergraphs},

url = {http://eudml.org/doc/270519},

volume = {27},

year = {2007},

}

TY - JOUR

AU - Yuejian Peng

TI - Subgraph densities in hypergraphs

JO - Discussiones Mathematicae Graph Theory

PY - 2007

VL - 27

IS - 2

SP - 281

EP - 297

AB - 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.

LA - eng

KW - Erdös jumping constant conjecture; Lagrangian; optimal vector

UR - http://eudml.org/doc/270519

ER -

## References

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- [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. Zbl0129.39902
- [10] Y. Peng, Non-jumping numbers for 4-uniform hypergraphs, Graphs and Combinatorics 23 (2007) 97-110, doi: 10.1007/s00373-006-0689-5. Zbl1115.05045
- [11] Y. Peng, Using Lagrangians of hypergraphs to find non-jumping numbers (I), submitted. Zbl1201.05101
- [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. Zbl1128.05029
- [13] J. Talbot, Lagrangians of hypergraphs, Combinatorics, Probability & Computing 11 (2002) 199-216, doi: 10.1017/S0963548301005053. Zbl0998.05049

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