Actually I got this question in job interview and successfully failed it ( it was obvious eliminating question ). I recall the Ramsey theory and cited the famous result on existence of cliques with edges of same color. But they then asked me to discuss how to estimate the probability that in fully connected graph there is a clique where edge colors are all different? Namely, they said: "Imagine that in a fully-connected graph, each edge is painted with one of two colours. Estimate the probability that the graph contains a fully-connected subgraph on 3 vertices that has edges of different colours"
I read a lot of articles but it does not look as naive result, which anyone knows since college graduation. So any help/hints/links are highly appreciated.
Thank you!
If you're looking for a triangle where all colors are different, then you obviously don't have enough different colors for that when there are only $2$ colors, so the probability is $0$.
If you're looking for a triangle where both colors are represented, then such a triangle will automatically exists unless all edges in the original graph have the same color.
Namely, suppose that $(v_1,v_2)$ is black and $(v_3,v_4)$ is white. If $(v_2,v_3)$ is black, then $\{v_2,v_3,v_4\}$ is a triangle that has both a black and a white edge; otherwise $\{v_1,v_2,v_3\}$ is such a triangle. (And if the two edges already share a vertex, just $\{v_1,v_2,v_3,v_4\}$ will do).
Thus the probability we're looking for is $$ 1 - 2^{-\binom{n}{2}+1} $$ which, when $n$ is not very small, will be so close to $1$ as not to matter.
Things get more involved with different numbers than $2$ colors and $3$-cliques, but it doesn't sound like that was what you were asked.