Skip to main content
×
×
Home

Distance Preserving Ramsey Graphs

  • DOMINGOS DELLAMONICA (a1) and VOJTĚCH RÖDL (a1)
Abstract

We prove the following metric Ramsey theorem. For any connected graph G endowed with a linear order on its vertex set, there exists a graph R such that in every colouring of the t-sets of vertices of R it is possible to find a copy G* of G inside R satisfying:

distG*(x, y) = distR(x, y) for every x, yV(G*);

the colour of each t-set in G* depends only on the graph-distance metric induced in G by the ordered t-set.

Copyright
References
Hide All
[1]Abramson, F. G. and Harrington, L. A. (1978) Models without indiscernibles. J. Symbolic Logic 43 572600.
[2]Deuber, W. (1975) Generalizations of Ramsey's theorem. In Infinite and Finite Sets I: Colloq., Keszthely, 1973, dedicated to P. Erdős on his 60th birthday, Vol. 10 of Colloq. Math. Soc. János Bolyai, North-Holland, pp. 323332.
[3]Deuber, W. (1975) Partitionstheoreme für Graphen. Comment. Math. Helv. 50 311320.
[4]Erdős, P., Hajnal, A. and Pósa, L. (1975) Strong embeddings of graphs into colored graphs. In Infinite and Finite Sets I: Colloq., Keszthely, 1973, dedicated to P. Erdős on his 60th birthday, Vol. 10 of Colloq. Math. Soc. János Bolyai, North-Holland, pp. 585595.
[5]Graham, R. L., Grötschel, M. and Lovász, L., editors (1995) Handbook of Combinatorics, Vols 1, 2, Elsevier Science.
[6]Graham, R. L., Leeb, K. and Rothschild, B. L. (1972) Ramsey's theorem for a class of categories. Proc. Nat. Acad. Sci. USA 69 119120.
[7]Graham, R. L., Rothschild, B. L. and Spencer, J. H. (1990) Ramsey Theory, second edition, Wiley-Interscience Series in Discrete Mathematics and Optimization.
[8]Graham, R. L. and Winkler, P. M. (1984) Isometric embeddings of graphs. Proc. Nat. Acad. Sci. USA 81 72597260.
[9]Hales, A. W. and Jewett, R. I. (1963) Regularity and positional games. Trans. Amer. Math. Soc. 106 222229.
[10]Nešetřil, J. (2005) Ramsey classes and homogeneous structures. Combin. Probab. Comput. 14 171189.
[11]Nešetřil, J. (2007) Metric spaces are Ramsey. European J. Combin. 28 457468.
[12]Nešetřil, J. and Rödl, V. (1975) Partitions of subgraphs. In Recent Advances in Graph Theory: Proc. Second Czechoslovak Sympos., Prague, 1974, Academia, pp. 413423.
[13]Nešetřil, J. and Rödl, V. (1977) Partitions of finite relational and set systems. J. Combin. Theory Ser. A 22 289312.
[14]Nešetřil, J. and Rödl, V. (1978) On a probabilistic graph-theoretical method. Proc. Amer. Math. Soc. 72 417421.
[15]Nešetřil, J. and Rödl, V. (1979) Partition theory and its application. In Surveys in Combinatorics: Proc. Seventh British Combinatorial Conf., Cambridge, 1979, Vol. 38 of London Math. Soc. Lecture Note Ser., Cambridge University Press, pp. 96156.
[16]Nešetřil, J. and Rödl, V. (1979) A short proof of the existence of highly chromatic hypergraphs without short cycles. J. Combin. Theory Ser. B 27 225227.
[17]Nešetřil, J. and Rödl, V. (1984) Combinatorial partitions of finite posets and lattices: Ramsey lattices. Algebra Universalis 19 106119.
[18]Nešetřil, J. and Rödl, V. (1987) Strong Ramsey theorems for Steiner systems. Trans. Amer. Math. Soc. 303 183192.
[19]Nešetřil, J. and Rödl, V. (1989) The partite construction and Ramsey set systems. Discrete Math. 75 327334.
[20]Nešetřil, J. and Rödl, V. (1990) Introduction: Ramsey theory old and new. In Mathematics of Ramsey Theory, Vol. 5 of Algorithms Combin., Springer, pp. 19.
[21]Nešetřil, J. and Rödl, V. (1992) On Ramsey graphs without bipartite subgraphs. Discrete Math. 101 223229.
[22]Rödl, V. (1973) A generalization of Ramsey theorem and dimension of graphs. Master's thesis, Charles University, Prague.
[23]Rödl, V. (1976) A generalization of the Ramsey theorem. In Graphs, Hypergraphs and Block Systems: Zielona Góra 1976, pp. 211–219.
[24]Winkler, P. M. (1984) Isometric embedding in products of complete graphs. Discrete Appl. Math. 7 221225.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Combinatorics, Probability and Computing
  • ISSN: 0963-5483
  • EISSN: 1469-2163
  • URL: /core/journals/combinatorics-probability-and-computing
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Keywords

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed