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Let $n ≥7$, and let $C$ be a collection of 15 distinct 5-element subsets of [n]. Prove that it is possible to color each element of $[n]$ yellow or orange so that each set belonging to $C$ has elements of both colors.

I know this is an applicaiton of Ramsey theory, but I'm not sure how to approach it since we could pick any n more than 7.

Maybe I'm completely wrong and it might not be ramsey theory related!

Also, I'm not sure how we'd relate this to a graph since we're coloring elements (vertices) as opposed to the edges themselves.

Thank you.

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  • $\begingroup$ You don't have to consider all of the $n$-set. You can ignore anything that isn't in one of the 15 sets in $C$. S $\endgroup$ – MJD Nov 27 '19 at 12:16
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Colour the elements randomly (independently and uniformly). The expected number of monochromatic subsets is $\frac{15}{16}\lt1$. Hence at least one colouring must be contributing to this expectation with $0$ monochromatic sets.

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  • $\begingroup$ How did you get 15/16 sorry? $\endgroup$ – thomas_stafford_22 Nov 27 '19 at 12:07
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    $\begingroup$ Each set can be colored in 32 ways, of which 2 are monochromatic. A randomly-colored set is monochromatic with probability $\frac2{32}=\frac1{16}$. You have 15 such sets, so the expected number of monochomatic sets is$\frac{15}{16}$. $\endgroup$ – MJD Nov 27 '19 at 12:23
  • $\begingroup$ So how does it being less than 1 mean that it's possible sorry? $\endgroup$ – thomas_stafford_22 Nov 27 '19 at 12:32

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