CHARACTERIZATION OF FINITE COLORED SPACES WITH CERTAIN CONDITIONS

A colored space is a pair (X, r) of a set X and a function r whose domain is (X 2). Let (X, r) be a finite colored space and Y, Z subset of X. We shall write Y similar or equal to(r) Z if there exists a bijection f : Y -> Z such that r(U) = r(f(U)) for each U is an element of (Y 2) where f(U) = {f(u) vertical bar u is an element of U}. We denote the numbers of equivalence classes with respect to similar or equal to(r) contained in (X i) by a(i)(r). In this paper we prove that a(2)(r) <= a(3)(r) when 5 <= vertical bar X vertical bar, and show what happens when equality holds.