# Subgraph densities in hypergraphs

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

top

## Abstract

top
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.

## How to cite

top

Yuejian 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

top
1. [1] D.P. Bertsekas, Constrained Optimization and Lagrange Multiplier Methods (Academic Press, New York, NY, 1982). Zbl0572.90067
2. [2] P. Erdös, On extremal problems of graphs and generalized graphs, Israel J. Math. 2 (1964) 183-190, doi: 10.1007/BF02759942. Zbl0129.39905
3. [3] P. Erdös and M. Simonovits, A limit theorem in graph theory, Studia Sci. Mat. Hung. Acad. 1 (1966) 51-57.
4. [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. Zbl0063.01277
5. [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. Zbl0731.05030
6. [6] P. Frankl and V. Rödl, Hypergraphs do not jump, Combinatorica 4 (1984) 149-159, doi: 10.1007/BF02579215. Zbl0663.05047
7. [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. Zbl1110.05052
8. [8] G. Katona, T. Nemetz and M. Simonovits, On a graph problem of Turán, Mat. Lapok 15 (1964) 228-238. Zbl0138.19402
9. [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. [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. [11] Y. Peng, Using Lagrangians of hypergraphs to find non-jumping numbers (I), submitted. Zbl1201.05101
12. [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. [13] J. Talbot, Lagrangians of hypergraphs, Combinatorics, Probability & Computing 11 (2002) 199-216, doi: 10.1017/S0963548301005053. Zbl0998.05049

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.