Characterization of finite colored spaces with certain conditions

A colored space is the pair $(X,r)$ of a set $X$ and a function $r$ whose domain is $\left(\genfrac{}{}{0px}{}{X}{2}\right)$. 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 $U\in \left(\genfrac{}{}{0px}{}{Y}{2}\right)$. We denote the numbers of equivalence classes with respect to ${\simeq }_r$ contained in $\left(\genfrac{}{}{0px}{}{X}{2}\right)$ and $\left(\genfrac{}{}{0px}{}{X}{3}\right)$ by $a_2(r)$ and $a_3(r)$, respectively. In this paper we prove that $a_2(r)\le a_3(r)$ when $5\le \mathrm{\mid }X\mathrm{\mid }$, and show what happens when the equality holds.