= Homogeneous set for a colouring
{title2=$\chi\restriction[H]^r\text{ is constant}$}
= Homogeneous sets for a colouring
{synonym}
For a colouring $\chi:[S]^r\to\{0,\ldots,k-1\}$ of the $r$-element subsets of a <set> $S$, a subset $H\subseteq S$ is homogeneous if all its $r$-element subsets receive the same colour. The infinite <Ramsey theorem for r-sets> guarantees an infinite homogeneous subset when $S=\mathbb N$, $r$ is positive and the number of colours is finite. This does not guarantee a <computable set> when the colouring is computable.
Back to article page
