Todorcevic's Open Coloring Axiom (OCA_T) states that any open graph on a separable metric space is either countably chromatic, or admits an uncountable clique. OCA_T has many interesting and important applications. Its known consistency proofs all lead to models where the continuum is $\aleph_2$. It is therefore natural to ask whether it implies that the continuum is $\aleph_2$, or whether there are other consistency proofs leading to models with larger continuum. (OCA_T negates the CH.) This question is still open. However we show that the restriction of OCA_T to spaces of size less than the continuum is consistent with arbitrarily large values of the continuum. Earlier work by Farah obtained this for the restriction to spaces of size $\aleph_1$.