Infinitary combinatorics

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In mathematics, infinitary combinatorics, or combinatorial set theory, is an extension of ideas in combinatorics to infinite sets. Some of the things studied include trees, extensions of Ramsey's theorem, and Martin's axiom.

[edit] Ramsey theory for infinite sets

Write κ, λ, m and n for cardinal numbers. The notation

\kappa\rightarrow(\lambda)^n_m

is a shorthand way of saying that every partition of the set [κ]n of n-element subsets of κ into m pieces has a homogeneous set of size λ. When m is 2 it is often omitted.

Some properties of this include:

\alef_0\rightarrow(\alef_0)^n_k for all finite n and k (Ramsey's theorem).
\beth_n^+\rightarrow(\alef_1)_{\alef_0}^{n+1} (Erdos-Rado theorem.)
2^\kappa\not\rightarrow(\kappa^+)^2
2^\kappa\not\rightarrow(3)^2_\kappa
\kappa\rightarrow(\kappa,\alef_0)^2 (Erdős-Dushnik-Miller theorem).

The uncountable cardinals κ such that κ→(κ)2 are the weakly compact cardinals.

[edit] References