Abstract
Show that in a party of six people there is always a group of three who either all know each other or are all strangers to each other. This well known puzzle is a special case of a theorem proved by Ramsey in 1928. The theorem has many deep extensions which are important not only in graph theory and combinatorics but in set theory (logic) and analysis as well. In this chapter we prove the original theorems of Ramsey, indicate some variations and present some applications of the results.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Notes
The fundamental Ramsey theorems are in F. P. Ramsey, On a problem of formal logic, Proc. Lond. Math. Soc. (2) 30 (1930) 264–286.
Theorems 4, 7 and 8 are due to V. Chvátal, P. Erdös and J. Spencer and can be found in S. A. Burr, Generalized Ramsey theory for graphs—a survey, in Graphs and Combinatorics (R. Bari and F. Harary, eds.) Springer-Verlag, 1974, pp. 52–75. For some of the deep results of Nešetřil and Rödl, one of which is Theorem 9,
see J. Nešetřil and V. Rödl, The Ramsey property for graphs with forbidden subgraphs, J. Combinatorial Theory Ser. B, 20 (1976) 243–249, and Partitions of finite relational and set systems, J. Combinatorial Theory Ser. A 22 (1977) 289–312.
The geometric Ramsey results of §3 can be found in P. Erdös, R. L. Graham, B. L. Rothschild, J. Spencer and E. G. Straus, Euclidean Ramsey theorems, I, J. Combinatorial Theory Ser. A, 14 (1973) 344–363.
Theorem 15 is in A. Hales and R. I. Jewett, Regularity and positional games, Trans. Amer. Math. Soc. 106 (1963) 222–229.
Rota’s conjecture is proved in R. L. Graham and B. L. Rothschild, Ramsey’s theorem for n-parameter sets, Trans. Amer. Math. Soc. 159 (1971) 257–292.
The results concerning Ramsey categories are given in R. L. Graham, K. Leeb and B. L. Rothschild, Ramsey’s theorem for a class of categories, Adv. in Math. 8 (1972) 417–433
Errata, R. L. Graham, K. Leeb and B. L. Rothschild, Ramsey’s theorem for a class of categories, Adv. in Math. 10 (1973) 326–327.
Van der Waerden’s theorem is in B. L. van der Waerden, Beweis einer Baudetschen Vermutung, Nieuw Archief voor Wiskunde 15 (1927) 212–216.
The simple proof we give is from R. L. Graham and B. L. Rothschild, A short proof of van der Waerden’s theorem on arithmetic progressions, Proc. Amer. Math. Soc. 42 (1974) 385–386.
Rado’s theorem, mentioned at the end of §3 is from R. Rado, Studien zur Kombinatorik, Math. Zeitschrift 36 (1933) 424–480.
The first results of §4 are in F. Galvin and K. Prikry, Borel sets and Ramsey’s theorem, J. Symbolic Logic 38 (1973) 193–198;
Glazer’s proof of Hindman’ theorem is in W. W. Comfort, Ultrafilters: some old and some new results, Bull. Amer. Math. Soc. 83 (1977) 417–455.
Exercise 5 is in R. E. Greenwood and A. M. Gleason, Combinatorial relations and chromatic graphs, Canad. J. Math. 7 (1955) 1–7,
Exercise 9 in a remark of F. Galvin and Schur’s theorem (Exercise 24) is in I. Schur, Über die Kongruenz x m + y m ≡ z m (mod p), Jahr. Deutsch. Math. Ver. 25 (1916)114–116.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1979 Springer-Verlag New York Inc.
About this chapter
Cite this chapter
Bollobás, B. (1979). Ramsey Theory. In: Graph Theory. Graduate Texts in Mathematics, vol 63. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-9967-7_6
Download citation
DOI: https://doi.org/10.1007/978-1-4612-9967-7_6
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4612-9969-1
Online ISBN: 978-1-4612-9967-7
eBook Packages: Springer Book Archive