The Ramsey property for C p (X)

For a Tychonoff space X, we denote by C _p (X) the space of real-valued continuous functions with the topology of pointwise convergence. We show that (a) Arhangel℉skii℉s property (α ₂) and the Ramsey property introduced by Nogura and Shakhmatov are equivalent for C _p (X), (b) the Ramsey property and Nyikos’ property (α _3/2) are not equivalent for C _p (X). These results answer questions posed by Shakhmatov. Concerning properties (α _i ) for C _p (X), some results on Scheepers’ conjecture are also given.