For an $n$-element metric space $X=\{x_1,\dots,x_n\}$ with metric $d$ we introduce an array containing $\frac{n(n-1)}2$ numbers $d(x_i,x_j)$, $i<j$. We assume that all distances are at least $1$. The number of relevant scales for the metric space $X$ is defined as the number of intervals $[2^{i-1},2^{i})$ $i=1,2,\dots$ containing some elements of the array $d(x_i,x_j)$, $i<j$. Let $RS(n)$ be the maximal number of relevant scales which an $n$-element metric space may have.

Problems: (1) What is the rate of growth of $RS(n)$? (2) What about exact evaluation of $RS(n)$ for all $n\in \mathbb{N}$?

Comments: (1) In Exercise 3.37 of my book "Metric Embeddings" I suggest readers to prove the following two statements as exercises (hints are given):

(i) $RS(n)\ge 2(n-1)$.

(ii) $RS(n)\le \frac{n(n-1)}2-(n-4)$.

So for nontrivially large $n$ the number $RS(n)$ is strictly between $n$ and $\frac{n(n-1)}2$.

(2) One can replace $2$ by another number in the definition of $RS(n)$, and it may change the answer.

For an $n$-element metric space $X=\{x_1,\dots,x_n\}$ with metric $d$ we introduce an array containing $\frac{n(n-1)}2$ numbers $d(x_i,x_j)$, $i<j$. We assume that all distances are at least $1$. The number of relevant scales for the metric space $X$ is defined as the number of intervals $[2^{i-1},2^{i})$ $i=1,2,\dots$ containing some elements of the array $d(x_i,x_j)$, $i<j$. Let $RS(n)$ be the maximal number of relevant scales which an $n$-element metric space may have.

Problems: (1) What is the rate of growth of $RS(n)$? (2) What about exact evaluation of $RS(n)$ for all $n\in \mathbb{N}$?

Comments: (1) In Exercise 3.37 of my book "Metric Embeddings" I suggest readers to prove the following two statements as exercises (hints are given):

(i) $RS(n)\ge 2n-3$.

(ii) $RS(n)\le \frac{n(n-1)}2-(n-4)$.

So for nontrivially large $n$ the number $RS(n)$ is strictly between $n$ and $\frac{n(n-1)}2$.

(2) One can replace $2$ by another number in the definition of $RS(n)$, and it may change the answer.

For an $n$-element metric space $X=\{x_1,\dots,x_n\}$ with metric $d$ we introduce an array containing $\frac{n(n-1)}2$ numbers $d(x_i,x_j)$, $i<j$. We assume that all distances are at least $1$. The number of relevant scales for the metric space $X$ is defined as the number of intervals $[2^{i-1},2^{i})$ $i=1,2,\dots$ containing some elements of the array $d(x_i,x_j)$, $i<j$. Let $RS(n)$ be the maximal number of relevant scales which an $n$-element metric space may have.

Problems: (1) What is the rate of growth of $RS(n)$? (2) What about exact evaluation of $RS(n)$ for all $n\in \mathbb{N}$?

Comments: (1) In Exercise 3.37 of my book "Metric Embeddings" I suggest readers to prove the following two statements as exercises (hints are given):

(i) $RS(n)\ge 2(n-1)$.

(ii) $RS(n)< \frac{n(n-1)}2-(n-4)$.

So for nontrivially large $n$ the number $RS(n)$ is strictly between $n$ and $\frac{n(n-1)}2$.

(2) One can replace $2$ by another number in the definition of $RS(n)$, and it may change the answer.

For an $n$-element metric space $X=\{x_1,\dots,x_n\}$ with metric $d$ we introduce an array containing $\frac{n(n-1)}2$ numbers $d(x_i,x_j)$, $i<j$. We assume that all distances are at least $1$. The number of relevant scales for the metric space $X$ is defined as the number of intervals $[2^{i-1},2^{i})$ $i=1,2,\dots$ containing some elements of the array $d(x_i,x_j)$, $i<j$. Let $RS(n)$ be the maximal number of relevant scales which an $n$-element metric space may have.

Problems: (1) What is the rate of growth of $RS(n)$? (2) What about exact evaluation of $RS(n)$ for all $n\in \mathbb{N}$?

Comments: (1) In Exercise 3.37 of my book "Metric Embeddings" I suggest readers to prove the following two statements as exercises (hints are given):

(i) $RS(n)\ge 2(n-1)$.

(ii) $RS(n)\le \frac{n(n-1)}2-(n-4)$.

So for nontrivially large $n$ the number $RS(n)$ is strictly between $n$ and $\frac{n(n-1)}2$.

(2) One can replace $2$ by another number in the definition of $RS(n)$, and it may change the answer.