K-distance set

Let $X$ be a set with metric $d$, $Y\subseteq X$, and $L=\{d(x,y):x,y\in Y,x\ne y\}$. If $K:=\mathrm{\#}(L)$ is finite, $Y$ is said to be a $K$-distance set.

$Y$ is called a maximal $K$-distance set if and only if for all $x\in X\setminus Y$, there exists $y\in Y$ such that $d(x,y)\notin L$. That is, if anything is added to $Y$, it is no longer a $K$-distance set.

$Y$ is called a spherical $K$-distance set if and only if $Y$ is a $K$-distance set and every element of $Y$ is a fixed distance $r$ from some element $c$, so $Y$ is a subset of the sphere (http://planetmath.org/SphereMetricSpace) centered at $c$ with radius $r$.

For example, let $X={ℝ}^2$ with $d=$ the box metric: $d(x,y)=\max \{|x_1-y_1|,|x_2-y_2|\}$ with $x_i,y_i$ components of $x,y$, respectively. Let $Y=\{(0,0),(1,0),(2,0),(0,1),(1,1),(2,1),(0,2),(1,2),(2,2)\}$. Then $L=\{1,2\}$, so $K=2$, so $Y$ is a 2-distance set.

Note: please do not confuse this definition of $K$-distance set with ${\mathrm{\Delta }}_K(Y)$, the $K$-distance set of $Y$.