Erdős–Rado theorem
Appearance
In
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).