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Could anyone explain me how to prove this:

We have $2$ colors. Red and Blue. If we color in a complete graph $K_n$ random edges with random colors, then there exists spanning tree, which contains only edges with given colors.

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    @BrianM.Scott I meant spanning tree. I have edited the post. – 2017-01-06
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    I suspect the problem is to show there exists a spanning tree which contains only edges *of a single color*. The mention of "random edges with random colors" is somewhat misleading. – 2017-01-06

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HINT: One way is to prove it by induction on $n$. Assume that the result is true for $n$, and consider $K_{n+1}$. Let $v$ be a a vertex of $K_{n+1}$, and let $E_v$ be the set of edges of $K_{n+1}$ having $v$ as an endpoint. Consider two cases:

  • $E$ is monochromatic.
  • $E$ contains edges of both colors. This is where you’ll use the induction hypothesis.