Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-26T18:56:19.845Z Has data issue: false hasContentIssue false
  • English
  • Français

Hypergraph Packing and Sparse Bipartite Ramsey Numbers

Published online by Cambridge University Press:  22 June 2009

DAVID CONLON
DAVID CONLON*
Affiliation:
St John's College, Cambridge CB2 1TP, UK (e-mail: [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove that there exists a constant c such that, for any integer Δ, the Ramsey number of a bipartite graph on n vertices with maximum degree Δ is less than 2cΔn. A probabilistic argument due to Graham, Rödl and Ruciński implies that this result is essentially sharp, up to the constant c in the exponent. Our proof hinges upon a quantitative form of a hypergraph packing result of Rödl, Ruciński and Taraz.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 18 , Issue 6 , November 2009 , pp. 913 - 923
Copyright
Copyright © Cambridge University Press 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Alon, N. (1994) Subdivided graphs have linear Ramsey numbers. J. Graph Theory 18 343–347.CrossRefGoogle Scholar
[2]Alon, N., Krivelevich, M. and Sudakov, B. (2003) Turán numbers of bipartite graphs and related Ramsey-type questions. Combin. Probab. Comput. 12 477–494.CrossRefGoogle Scholar
[3]Beck, J. (1983) An upper bound for diagonal Ramsey numbers. Studia Sci. Math. Hungar. 18 401–406.Google Scholar
[4]Burr, S. A. and Erdős, P. (1975) On the magnitude of generalized Ramsey numbers for graphs. In Infinite and Finite Combinatorics 1, Vol. 10 of Colloq. Math. Soc. János Bolyai, pp. 214–240.Google Scholar
[5]Chen, G. and Schelp, R. (1993) Graphs with linearly bounded Ramsey numbers. J. Combin. Theory Ser. B 57 138–149.CrossRefGoogle Scholar
[6]Chung, F. and Graham, R. L. (1998) Erdős on Graphs: His Legacy of Unsolved Problems, A. K. Peters, Wellesley, MA.CrossRefGoogle Scholar
[7]Chvátal, V., Rödl, V., Szemerédi, E. and Trotter, W. T. Jr., (1983) The Ramsey number of a graph with bounded maximum degree. J. Combin. Theory Ser. B 34 239–243.CrossRefGoogle Scholar
[8]Eaton, N. (1998) Ramsey numbers for sparse graphs. Discrete Math. 185 63–75.CrossRefGoogle Scholar
[9]Erdős, P. (1984) On some problems in graph theory, combinatorial analysis and combinatorial number theory. In Graph Theory and Combinatorics (Cambridge 1983), Academic Press, London, pp. 1–17.Google Scholar
[10]Erdős, P. and Szekeres, G. (1935) A combinatorial problem in geometry. Compositio Mathematica 2 463–470.Google Scholar
[11]Fox, J. and Sudakov, B. Density theorems for bipartite graphs and related Ramsey-type results. Combinatorica, to appear.Google Scholar
[12]Gowers, W. T. (1997) Lower bounds of tower type for Szemerédi's uniformity lemma. Geom. Funct. Anal. 7 322–337.CrossRefGoogle Scholar
[13]Gowers, W. T. (1998) A new proof of Szemerédi's theorem for arithmetic progressions of length 4. Geom. Funct. Anal. 8 529–551.CrossRefGoogle Scholar
[14]Graham, R. L., Rödl, V. and Ruciński, A. (2000) On graphs with linear Ramsey numbers. J. Graph Theory 35 176–192.3.0.CO;2-C>CrossRefGoogle Scholar
[15]Graham, R. L., Rödl, V. and Ruciński, A. (2001) On bipartite graphs with linear Ramsey numbers. Combinatorica 21 199–209.CrossRefGoogle Scholar
[16]Komlós, J. and Simonovits, M. (1996) Szemerédi's regularity lemma and its applications in graph theory. In Combinatorics: Paul Erdős is Eighty, Vol. 2 (Keszthely 1993), Budapest, pp. 295–352.Google Scholar
[17]Kostochka, A. and Rödl, V. (2001) On graphs with small Ramsey numbers. J. Graph Theory 37 198–204.CrossRefGoogle Scholar
[18]Kostochka, A. and Rödl, V. (2004) On graphs with small Ramsey numbers II. Combinatorica 24 389–401.CrossRefGoogle Scholar
[19]Kostochka, A. and Sudakov, B. (2003) On Ramsey numbers of sparse graphs. Combin. Probab. Comput. 12 627–641.CrossRefGoogle Scholar
[20]Ramsey, F. P. (1930) On a problem of formal logic. Proc. London Math. Soc. Ser. 2 30 264–286.CrossRefGoogle Scholar
[21]Rödl, V., Ruciński, A. and Taraz, A. (1999) Hypergraph packing and graph embedding. Combin. Probab. Comput. 8 363–376.CrossRefGoogle Scholar
[22]Sauer, N. and Spencer, J. (1978) Edge disjoint placement of graphs. J. Combin. Theory Ser. B 25 295–302.CrossRefGoogle Scholar
[23]Shi, L. (2001) Cube Ramsey numbers are polynomial. Random Struct. Alg. 19 99–101.Google Scholar
[24]Shi, L. (2007) The tail is cut for Ramsey numbers of cubes. Discrete Math. 307 290–292.CrossRefGoogle Scholar
[25]Sudakov, B. (2003) A few remarks on Ramsey–Turán-type problems. J. Combin. Theory Ser. B 88 99–106.CrossRefGoogle Scholar