Théorème d'Erdős – Dushnik – Miller

Erdős–Dushnik–Miller theorem Not to be confused with Dushnik–Miller theorem.

In the mathematical theory of infinite graphs, the Erdős–Dushnik–Miller theorem is a form of Ramsey's theorem stating that every infinite graph contains either a countably infinite independent set, or a clique with the same cardinality as the whole graph.[1] The theorem was first published by Ben Dushnik and E. O. Miller (1941), in both the form stated above and an equivalent complementary form: every infinite graph contains either a countably infinite clique or an independent set with equal cardinality to the whole graph. In their paper, they credited Paul Erdős with assistance in its proof. They applied these results to the comparability graphs of partially ordered sets to show that each partial order contains either a countably infinite antichain or a chain of cardinality equal to the whole order, and that each partial order contains either a countably infinite chain or an antichain of cardinality equal to the whole order.[2] The same theorem can also be stated as a result in set theory, using the arrow notation of Erdős & Rado (1956), comme {displaystyle kappa rightarrow (kappa ,aleph _{0})^{2}} . Cela signifie que, for every set {style d'affichage S} of cardinality {style d'affichage kappa } , and every partition of the ordered pairs of elements of {style d'affichage S} into two subsets {style d'affichage P_{1}} et {style d'affichage P_{2}} , there exists either a subset {style d'affichage S_{1}subset S} of cardinality {style d'affichage kappa } or a subset {style d'affichage S_{2}subset S} of cardinality {displaystyle aleph _{0}} , such that all pairs of elements of {style d'affichage S_{je}} belong to {style d'affichage P_{je}} .[3] Ici, {style d'affichage P_{1}} can be interpreted as the edges of a graph having {style d'affichage S} as its vertex set, in which {style d'affichage S_{1}} (if it exists) is a clique of cardinality {style d'affichage kappa } , et {style d'affichage S_{2}} (if it exists) is a countably infinite independent set.[1] Si {style d'affichage S} is taken to be the cardinal number {style d'affichage kappa } lui-même, the theorem can be formulated in terms of ordinal numbers with the notation {displaystyle kappa rightarrow (kappa ,oméga )^{2}} , qui veut dire {style d'affichage S_{2}} (when it exists) has order type {style d'affichage oméga } . For uncountable regular cardinals {style d'affichage kappa } (and some other cardinals) this can be strengthened to {displaystyle kappa rightarrow (kappa ,oméga +1)^{2}} ;[4] toutefois, it is consistent that this strengthening does not hold for the cardinality of the continuum.[5] The Erdős–Dushnik–Miller theorem has been called the first "unbalanced" generalization of Ramsey's theorem, and Paul Erdős's first significant result in set theory.[6] Références ^ Aller à: a b Milner, E. C; Pouzet, M. (1985), "The Erdős–Dushnik–Miller theorem for topological graphs and orders", Order, 1 (3): 249–257, est ce que je:10.1007/BF00383601, M 0779390, S2CID 123272176; see in particular Theorem 44 ^ Dushnik, Ben; Miller, E. O. (1941), "Partially ordered sets", Journal américain de mathématiques, 63 (3): 600–610, est ce que je:10.2307/2371374, JSTOR 2371374, M 0004862; see in particular Theorems 5.22 et 5.23 ^ Forêt, Paul; Rado, R. (1956), "A partition calculus in set theory", Bulletin de l'American Mathematical Society, 62 (5): 427–489, est ce que je:10.1090/S0002-9904-1956-10036-0, M 0081864 ^ Shelah, Sahara (2009), "The Erdős–Rado arrow for singular cardinals", Canadian Mathematical Bulletin, 52 (1): 127–131, est ce que je:10.4153/CMB-2009-015-8, M 2494318 ^ Shelah, Sahara; Stanley, Lee J. (2000), "Filters, Cohen sets and consistent extensions of the Erdős–Dushnik–Miller theorem", Le Journal de la logique symbolique, 65 (1): 259–271, arXiv:math/9709228, est ce que je:10.2307/2586535, JSTOR 2586535, M 1782118, S2CID 2763013 ^ Hajnal, András (1997), "Paul Erdős' set theory", The mathematics of Paul Erdős, II, Algorithms and Combinatorics, volume. 14, Berlin: Springer, pp. 352–393, est ce que je:10.1007/978-3-642-60406-5_33, M 1425228; see in particular Section 3, "Infinite Ramsey theory – early papers", p. 353 Catégories: Order theoryRamsey theorySet theory