# Ramsey Theory - Applying to finite sets

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.

• 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 – MJD Nov 27 '19 at 12:16

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.
• 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$\frac{15}{16}$. – MJD Nov 27 '19 at 12:23