A note on balanced colourings for lattice points

Abstract

The following problem was posed in the 27th International Mathematics Olympiad (1986): One is given a finite set of points Pn in the plane, each point having integer coordinates. Is it always possible to colour some of the points red and the remaining points white in such a way that, for any straight line L parallel to either one of the coordinate axes, the difference (in absolute value) between the number of white points and red points on L is not greater than 1? It is not hard to see that the answer to the above question is "yes". In this note we generalize this result, and show that Pn can be coloured with m (m≥2) colours in such a way that for any straight line parallel to either one of the coordinate axes, the difference (in absolute value) between the number of points coloured i and the number of points coloured j is at most 1, 1≤i