For a Tychonoff space X, we denote by Cp(X) the space of real-valued continuous functions with the topology of pointwise convergence. We show that (a) Arhangel℉skii℉s
property (α2) and the Ramsey property introduced by Nogura and Shakhmatov are equivalent for Cp(X), (b) the Ramsey property and Nyikos’ property (α3/2) are not equivalent for Cp(X). These results answer questions posed by Shakhmatov. Concerning properties (αi) for Cp(X), some results on Scheepers’ conjecture are also given.