Journal of the
Korean Mathematical Society JKMS

A \textit{colored space} is {a pair} $(X,r)$ of a set $X$ and a function $r$ whose domain is $X\choose 2$. 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 $U\in {Y\choose 2}$ {where $f(U)=\{f(u)\mid u\in U\}$}. We denote the numbers of equivalence classes with respect to $\simeq_r$ contained in {$X \choose i$ by $a_i(r)$}. In this paper we prove that $a_2(r)\leq a_3(r)$ when $5\leq |X|$, and show what happens {when equality} holds.

Keywords: colored spaces, isometric sequences, distance sets

MSC numbers: 05C15, 05C35

Supported by: This work was supported by a 2-Year Research Grant of Pusan National University.