Erdős–Rado theorem

Template:For Template:Short description 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.Template:Sfnp It is sometimes also attributed to Đuro Kurepa who proved it under the additional assumption of the generalised continuum hypothesis,Template:Sfnp 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

${\exp }_r(\kappa {)}^{+}⟶({\kappa }^{+}{)}_{\kappa }^{r+1}$

where exp₀(κ) = κ and inductively exp_r+1(κ)=2^exp_r(κ). This is sharp in the sense that exp_r(κ)⁺ cannot be replaced by exp_r(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 exp_r(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).

References

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