Characterization of finite colored spaces with certain conditions

Abstract

A colored space is the pair $(X,r)$ of a set $X$ and a function $r$ whose domain is $\binom{X}{2}$. Let $(X,r)$ be a finite colored space and $Y,Z\subseteq X$. We shall write $Y\simeq_r Z$ if there exists a bijection $f:Y\to Z$ such that $r(U)=r(f(U))$ for each $U\in\binom{Y}{2}$. We denote the numbers of equivalence classes with respect to $\simeq_r$ contained in $\binom{X}{2}$ and $\binom{X}{3}$ by $a_2(r)$ and $a_3(r)$, respectively. In this paper we prove that $a_2(r)\leq a_3(r)$ when $5\leq |X|$, and show what happens when the equality holds.