Home > Combinatorics, Measure Theory, Problem Solving > Measurable set has nice property.

Measurable set has nice property.


Let S \subset [0,1]^2 be a measurable set such that m(S)>1/2 (in the Lebesgue measure). Prove that there exist x,y,z such that (x,y),(x,z),(y,z) \in S.

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  1. Marco Cioffi
    March 4, 2011 at 8:20 pm

    S is Lebesgue measurable iff for all e>0 exist a compact K_e subset of S that m(S-K_e)<e. We can be sure to a small enough 2-cube in S. Three of the vertex of this 2-cube are the points we want to find. My speak is very bad, sorry. Is the proof correct?

  2. March 4, 2011 at 8:25 pm

    If you notice, one of the vertices of the rectangle should be on the diagonal, so I’m afraid its more complicated than that.

  3. yan
    April 25, 2013 at 11:04 am

    I think I have a solution. Are you still interested or has it been solved by now ?

    • April 25, 2013 at 12:03 pm

      If you have a solution you are welcome to post it. 🙂

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