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Jan 29, 2014 at 7:36 comment added Andrew Ryzhikov Example: for three intervals $[0, 3]$, $[1, 4]$, $[2, 5]$ and $k = 2$ we need to colour first and third interval in first colour and second interval in second colour, so the measure of the set of points, which are contained in two or more intervals coloured with the same colour, is $1$.
Jan 29, 2014 at 7:19 comment added Andrew Ryzhikov so what? Measure of a point is zero. If $S$ is a finite set, the optimization problem is bounded.
Jan 29, 2014 at 2:28 comment added Alexandre Eremenko Still the problem does not make much sense to me: what if all your intervals contain a common point?
Jan 28, 2014 at 15:15 comment added Andrew Ryzhikov I am interested in this problem for finite set $S$, I have added it explicitly to the post.
Jan 28, 2014 at 14:32 history answered Alexandre Eremenko CC BY-SA 3.0