Let X be a finite set of points on the plane such that each point in X is colored with red or blue and there is no line having all points in X. Prove that there is a line L having at least two points of X such that all points in L∩X have the same color.
Let k be a postivive integer. Let f(k) be the minimum number n such that no matter how we color the integer points in {(x,y,z): 0<x,y,z≤n} with k colors, there always exist 8 monochromatic points forming the vertices of a box whose sides are parallel to xy- or yz- or xz- plane. Determine f(k).
Suppose that we color integers 1, 2, 3, …, n with three colors so that each color is given to more than n/4 integers. Prove that there exist x, y, z such that x+y=z and x,y,z have distinct colors.