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.[1] It is sometimes also attributed to Đuro Kurepa who proved it under the additional assumption of the generalised continuum hypothesis,[2] 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

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
  1. Erdős & Rado (1956).
  2. https://mathoverflow.net/questions/191326/where-is-the-erd%C5%91s-rado-theorem-stated-in-erd%C5%91s-and-rados-bull-ams-paper#comment495809_191327

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</ref>
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