An algebraic proof of a conjecture of Erdos and Purdy

Erdos and Purdy conjectured that for n \geq 5 one cannot find a set of n red points in general position and another set of n-1 blue points such that every line determined by two red points passes also through a blue point. This conjecture is trivially true for n odd, but turns to be very challenging for n even. The conjecture is, in fact, a special (but important) case of the Magic Configuration conjecture of Murty from 1971. The conjecture of Murty was proved in 2008 in a topological setting.

Here we present a purely algebraic proof of the conjecture of Erdos and Purdy. On the way we also provide a nice result on vectors in the two dimensional plane.