MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Suppose we have a finite, 100-uniform system of sets such that any point is contained in at most 3 sets. Is it true that we can color the points such that every set contains 50 red and 50 blue points?

The question is by Thomas Rothvoss. A positive answer would solve the Three permutations problem of Beck, so a simple answer would be a counterexample...

share|cite|improve this question
I'm sorry, what is a "100-uniform system of sets"? Is that a system of sets, each of which contains exactly 100 elements? If so, why 100? Is the problem trivial for 98 (with each set containing 49 of each color)? Uninteresting for 102? It's certainly false for 2; consider the sets a-b, a-c, b-c, a 2-uniform system such that no point is in more than 3 (or even 2) sets, yet no way to color the points so each set has a red and a blue. – Gerry Myerson Jun 23 '10 at 13:05
up vote 5 down vote accepted


Given sets $$ a_1,a_2,\dots,a_{99},b_1{\rm\ and\ }a_1,a_2,\dots,a_{99},b_2 $$ we see that $b_1$ and $b_2$ must be the same color, say, red. Then from $$ b_1,b_2,c_1,c_2,\dots,c_{98}{\rm\ and\ }d_1,d_2,c_1,c_2,\dots,c_{98} $$ we see $d_1$ and $d_2$ must both be red. Then from $$ d_1,d_2,e_1,e_2,\dots,e_{98}{\rm\ and\ }f_1,f_2,e_1,e_2,\dots,e_{98} $$ we see that $f_1$ and $f_2$ must both be red. Dot, dot, dot. You wind up with as many elements as you like, all of which must be red, and none of them are in more than two of the sets. Once you have more than 50 of them, you can put them in another set which will then have more than 50 red points.

Obviously, we can take 100 to be a variable in this problem and solution, provided we restrict its range to the positive even integers and understand 50 to be 100/2.

share|cite|improve this answer
Wow, this was simpler than I thought, thanks! – domotorp Jun 24 '10 at 11:10

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.