AbstractRamsey's theorem implies that every functionf:0, 1 ℝ isconvex or concave on an infinite set. We show that there is an upper semicontinuous function which is not convex or concave on any uncountable set. We investigate those functions which are not convex on any r element set (r). A typical result: iffis bounded from below and is not convex on any infiniteset then there exists an interval on which the graph offcan be covered by the graphs of countably many strictly concave functions.