  We introduce natural strengthenings of sequential compactness called the
$r$-Ramsey property for each natural number $r\geq 1$. We prove that metrizable
compact spaces are $r$-Ramsey for all $r$ and give examples of compact spaces
that are $r$-Ramsey but not $r+1$-Ramsey for each $r\geq 1$ (assuming CH for
all $r>1$
