Erdős–Rado theorem

Erdős–Rado theorem In partition calculus, part of combinatorial set theory, a branch of mathematics, the Erdős–Rado theorem is a basic result extending Ramsey's theorem to uncountable sets. It is named after Paul Erdős and Richard Rado. It is sometimes also attributed to Đuro Kurepa who proved it under the additional assumption of the generalised continuum hypothesis,[1] and hence the result is sometimes also referred to as the Erdős–Rado–Kurepa theorem.

Statement of the theorem If r ≥ 0 is finite and κ is an infinite cardinal, then {displaystyle exp _{r}(kappa )^{+}longrightarrow (kappa ^{+})_{kappa }^{r+1}} where exp0(κ) = κ and inductively expr+1(κ)=2expr(κ). This is sharp in the sense that expr(κ)+ cannot be replaced by expr(κ) on the left hand side.

The above partition symbol describes the following statement. If f is a coloring of the r+1-element subsets of a set of cardinality expr(κ)+, in κ many colors, then there is a homogeneous set of cardinality κ+ (a set, all whose r+1-element subsets get the same f-value).

Notes ^ References Erdős, Paul; Hajnal, András; Máté, Attila; Rado, Richard (1984), Combinatorial set theory: partition relations for cardinals, Studies in Logic and the Foundations of Mathematics, vol. 106, Amsterdam: North-Holland Publishing Co., ISBN 0-444-86157-2, MR 0795592 Erdős, P.; Rado, R. (1956), "A partition calculus in set theory.", Bull. Amer. Math. Soc., 62 (5): 427–489, doi:10.1090/S0002-9904-1956-10036-0, MR 0081864 Categories: Set theoryTheorems in combinatoricsPaul Erdős

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