Let n and k be positive integers and let
T={(x,y,z)∈N3∣1≤x,y,z≤n}
be the length n lattice cube. Suppose that 3n2−3n+1+k points of T are colored red such that if P and Q are red points and PQ is parallel to one of the coordinate axes, then the whole line segment PQ consists of only red points. Prove that there exists at least k unit cubes of length 1, all of whose vertices are colored red. 3D geometrycombinatoricslattice points