I am reading a paper and it mentions a continuous analogue of a related graph theory question that people concern. The question is that suppose $E\subset Q=[0,1]^2$ has lebesgue measure $|E|>0$, is it true that for some constant $\epsilon$ independent of $E$, one can always find four points $A, B,C,D\in E$ such that the quadrilateral figure $ABCD$ is a rectangle with sides parallel to the coordinate axes, whose area is at least $\epsilon|E|^2$? I guess it is still an open question. What I want to ask is why people care about this question? Does it have some applications in graph theory? Thanks for any reference and help!
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$\begingroup$ If you provided additional information about the related results which would justify the occurence of $\ \epsilon\!\cdot\! |E|^2\ $ then I would consider this question worthy simply by the virtue of its simplicity and unobviousness. $\endgroup$– Włodzimierz HolsztyńskiJun 29, 2016 at 21:34
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$\begingroup$ @piano what paper are you reading? $\endgroup$– TurboJun 29, 2016 at 21:45
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$\begingroup$ I don't know the reason for the occurence of $\epsilon |E|^2$...The paper just mentions the result. $\endgroup$– violinJun 29, 2016 at 21:46
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$\begingroup$ This problem seems to me to be about showing that the intersection of, say, four translates of a set of a positive measure is (or is not) empty. $\endgroup$– Włodzimierz HolsztyńskiJun 29, 2016 at 21:46
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$\begingroup$ @piano, do your homework, read the literature or think more about the problem, find out about $\ \epsilon\!\cdot\!|E|^2$. $\endgroup$– Włodzimierz HolsztyńskiJun 29, 2016 at 21:48
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