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 $binom{X}{2}$. Let $(X,r)$ be a finite colored space and $Y,Zsubseteq X$. We shall write $Ysimeq_r Z$ if there exists a bijection $f:Yto Z$ such that $r(U)=r(f(U))$ for each $Uinbinom{Y}{2}$. 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 $5leq |X|$, and show what happens when the equality holds.