Erdős–Dushnik–Miller theorem

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. C. 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), Como {displaystyle kappa rightarrow (capa ,aleph _{0})^{2}} . Isso significa que, for every set {estilo de exibição S} of cardinality {kappa de estilo de exibição } , and every partition of the ordered pairs of elements of {estilo de exibição S} into two subsets {estilo de exibição P_{1}} e {estilo de exibição P_{2}} , there exists either a subset {estilo de exibição S_{1}subset S} of cardinality {kappa de estilo de exibição } or a subset {estilo de exibição S_{2}subset S} of cardinality {displaystyle aleph _{0}} , such that all pairs of elements of {estilo de exibição S_{eu}} belong to {estilo de exibição P_{eu}} .[3] Aqui, {estilo de exibição P_{1}} can be interpreted as the edges of a graph having {estilo de exibição S} as its vertex set, in which {estilo de exibição S_{1}} (if it exists) is a clique of cardinality {kappa de estilo de exibição } , e {estilo de exibição S_{2}} (if it exists) is a countably infinite independent set.[1] Se {estilo de exibição S} is taken to be the cardinal number {kappa de estilo de exibição } em si, the theorem can be formulated in terms of ordinal numbers with the notation {displaystyle kappa rightarrow (capa ,ómega )^{2}} , significa que {estilo de exibição S_{2}} (when it exists) has order type {displaystyle ômega } . For uncountable regular cardinals {kappa de estilo de exibição } (and some other cardinals) this can be strengthened to {displaystyle kappa rightarrow (capa ,ómega +1)^{2}} ;[4] Contudo, 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] Referências ^ Ir para: a b Milner, E. C.; Pouzet, M. (1985), "The Erdős–Dushnik–Miller theorem for topological graphs and orders", Order, 1 (3): 249-257, doi:10.1007/BF00383601, SENHOR 0779390, S2CID 123272176; see in particular Theorem 44 ^ Dushnik, Ben; Miller, E. C. (1941), "Partially ordered sets", Revista Americana de Matemática, 63 (3): 600-610, doi:10.2307/2371374, JSTOR 2371374, SENHOR 0004862; see in particular Theorems 5.22 e 5.23 ^ Floresta, Paulo; Rado, R. (1956), "A partition calculus in set theory", Boletim da American Mathematical Society, 62 (5): 427–489, doi:10.1090/S0002-9904-1956-10036-0, SENHOR 0081864 ^ Shelah, Saharon (2009), "The Erdős–Rado arrow for singular cardinals", Canadian Mathematical Bulletin, 52 (1): 127–131, doi:10.4153/CMB-2009-015-8, SENHOR 2494318 ^ Shelah, Saharon; Stanley, Lee J. (2000), "Filters, Cohen sets and consistent extensions of the Erdős–Dushnik–Miller theorem", O Diário da Lógica Simbólica, 65 (1): 259-271, arXiv:math/9709228, doi:10.2307/2586535, JSTOR 2586535, SENHOR 1782118, S2CID 2763013 ^ Hajnal, András (1997), "Paul Erdős' set theory", The mathematics of Paul Erdős, II, Algorithms and Combinatorics, volume. 14, Berlim: Springer, pp. 352-393, doi:10.1007/978-3-642-60406-5_33, SENHOR 1425228; see in particular Section 3, "Infinite Ramsey theory – early papers", p. 353 Categorias: Order theoryRamsey theorySet theory