Abstract
Dedekind’s pigeon-hole principle, also known as the box argument or the chest of drawers argument (Schubfachprinzip) can be described, rather vaguely, as follows. If sufficiently many objects are distributed over not too many classes, then at least one class contains many of these objects. In 1930 F. P. Ramsey [12] discovered a remarkable extension of this principle which, in its simplest form, can be stated as follows. Let S be the set of all positive integers and suppose that all unordered pairs of distinct elements of S are distributed over two classes.
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References
F. Bagemihl and H. D. Sprinkle, On a proposition of Sierpińiski, Proc. Amer. Math. Soc. vol. 5 (1954) pp. 726-728.
B. Dushnik and E. W. Miller, Partially ordered sets, Amer. J. Math. vol. 63 (1941) p. 605.
P. Erdös, Some set-theoretical properties of graphs, Revista Universidad Nacional de Tucuman, Serie A vol. 3 (1942) pp. 363-367.
P. Erdös and R. Rado, A combinatorial theorem, J. London Math. Soc. vol. 25 (1950) pp. 249-255.
P. Erdös and R. Rado, Combinatorial theorems on classifications of subsets of a given set, Proc. London Math. Soc. (3) vol. 2 (1952) pp. 417-439.
P. Erdös and R. Rado, A problem on ordered sets, J. London Math. Soc. vol 28 (1953) pp. 426-438.
P. Erdös and G. Szekeres, A combinatorial problem in geometry, Compositio Math. vol. 2 (1935) pp. 463-470.
P. Hall, On representations of sub-sets, J. London Math. Soc. vol. 10 (1934) pp. 26-30.
F. Hausdorff, Grundztige einer theorie der geordneten Mengen, Math. Ann. vol. 65 (1908) pp. 435-506.
F. Hausdorff, Mengenlehre, 3d ed., 1944, §16.
R. Rado, Direct decompositions of partitions, J. London Math. Soc. vol. 29 (1954), pp. 71-83.
F. P. Ramsey, On a problem of formal logic, Proc. London Math. Soc. (2) vol. 30 (1930) pp. 264-286.
W. Sierpińiski, Lemons sur les nombres transfinis, Paris, 1928.
W. Sierpińiski, Ojednom problemu G Ruzjevića koji se odnosi na hipotezu kontinuuma, Glas Srpske Kraljevske Akademije vol. 152 (1932) pp. 163-169.
W. Sierpińiski, Sur un probléme de la théorie des relations, Annali R. Scuola Normale Superiore de Pisa Ser. 2 vol. 2 (1933) pp. 285-287.
W. Sierpińiski, Concernant l’hypothèse du continu, Académie Royale Serbe. Bulletin de l’Académie des Sciences Mathématiques et Naturelles. A. Sciences Mathématiques et Physiques vol. 1 (1933) pp. 67-73.
A. Tarski, Quelques théorèmes sur les alephs, Fund. Math. vol. 7 (1925) p. 2.
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Erdös, P., Rado, R. (2009). A Partition Calculus in Set Theory. In: Gessel, I., Rota, GC. (eds) Classic Papers in Combinatorics. Modern Birkhäuser Classics. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4842-8_14
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