Solovay has shown that if $F: \lbrack\omega\rbrack^\omega \rightarrow 2$ is a clopen partition with recursive code, then there is an infinite homogeneous hyperarithmetic set for the partition (a basis result). Simpson has shown that for every $0^\alpha$, where $\alpha$ is a recursive ordinal, there is a clopen partition $F: \lbrack\omega\rbrack^\omega \rightarrow 2$ such that every infinite homogeneous set is Turing above $0^\alpha$ (an anti-basis result). Here we refine these results, by associating the "order type" of a clopen set with the Turing complexity of the infinite homogeneous sets. We also consider the Nash-Williams barrier theorem and its relation to the clopen Ramsey theorem.