MathDB

Let

n

integers on the real axis be colored. Determine for which positive integers

k

there exists a family

K

of closed intervals with the following properties: i) The union of the intervals in

K

contains all of the colored points; ii) Any two distinct intervals in

K

are disjoint; iii) For each interval

I

K

we have

{{a}_{I}}=k.{{b}_{I}}

, where

{{a}_{I}}

denotes the number of integers in

I

, and

{{b}_{I}}

the number of colored integers in

I

Problem Statement