Ramsey's theorem and computability theory

An infinite version of Ramsey's theorem states that for every coloring
of

$[\omega]^n$ (the set of all $n$--element subsets of $\omega$) by finitely

many colors, there is an infinite set $A$ which is homogeneous for
that

coloring, i.e., all elements of $[A]^n$ have the same color. After
an

introduction to the necessary background in computability theory, we

present a survey of results concerning the complexity of infinite

homogeneous sets for effectively given (computable or computably

enumerable) colorings.