Abstract
The finite version of Ramsey’s theorem asserts that for every triple k, m and r of positive integers there exists a positive integer n such that n → (m) r k. In Sect. 1.2 a compactness argument was used to derive this result from the infinite Ramsey theorem not giving any information about the size of n
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References
Ajtai, M., Komlós, J., Szemerédi, E.: A note on Ramsey numbers. J. Comb. Theory A 29, 354–360 (1980)
Ajtai, M., Komlós, J., Szemerédi, E.: A dense infinite Sidon sequence. Eur. J. Comb. 2, 1–11 (1981)
Bohman, T.: The triangle-free process. Adv. Math. 221, 1653–1677 (2009)
Bohman, T., Keevash, P.: The early evolution of the H-free process. Invent. Math. 181, 291–336 (2010)
Bollobás, B.: Random Graphs, 2nd edn. Cambridge University Press, Cambridge (2001)
Chung, F.R.K., Grinstead, C.M.: A survey of bounds for classical Ramsey numbers. J. Graph Theory 7, 25–37 (1983)
Conlon, D.: A new upper bound for diagonal Ramsey numbers. Ann. Math. 170, 941–960 (2009)
Erdős, P.: Some remarks on the theory of graphs. Bull. Am. Math. Soc. 53, 292–294 (1947)
Erdős, P.: Graph theory and probability. II. Can. J. Math. 13, 346–352 (1961)
Erdős, P.: On the combinatorial problems which I would most like to see solved. Combinatorica 1, 25–42 (1981)
Erdős, P.: Welcoming address. In: Random Graphs ’83 (Poznań, 1983), pp. 1–5. North-Holland, Amsterdam (1985)
Erdős, P., Rado, R.: A combinatorial theorem. J. Lond. Math. Soc. 25, 249–255 (1950)
Erdős, P., Rado, R.: Combinatorial theorems on classifications of subsets of a given set. Proc. Lond. Math. Soc. 2, 417–439 (1952)
Erdős, P., Szekeres, G.: A combinatorial problem in geometry. Compos. Math. 2, 463–470 (1935)
Fredricksen, H.: Schur numbers and the Ramsey numbers N(3, 3, ⋯, 3; 2). J. Comb. Theory A 27, 376–377 (1979)
Graver, J.E., Yackel, J.: Some graph theoretic results associated with Ramsey’s theorem. J. Comb. Theory 4, 125–175 (1968)
Greenwood, R.E., Gleason, A.M.: Combinatorial relations and chromatic graphs. Can. J. Math. 7, 1–7 (1955)
Kalbfleisch, J.G.: Chromatic graphs and Ramsey’s theorem. Phd. Thesis, University of Waterloo (1966)
Kim, J.H.: The Ramsey number R(3, t) has order of magnitude t 2∕logt. Random Struct. Algorithms 7, 173–207 (1995)
Radziszowski, S.: Small Ramsey numbers. Electron. J. Comb. Dyn. Surv. DS1 (2011)
Shearer, J.B.: A note on the independence number of triangle-free graphs. Discrete Math. 46, 83–87 (1983)
Spencer, J.: Ramsey’s theorem – a new lower bound. J. Comb. Theory A 18, 108–115 (1975a)
Spencer, J.: Asymptotic lower bounds for Ramsey functions. Discrete Math. 20, 69–76 (1977)
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Prömel, H.J. (2013). Ramsey Numbers. In: Ramsey Theory for Discrete Structures. Springer, Cham. https://doi.org/10.1007/978-3-319-01315-2_7
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