Can we prove or disprove the following statement?
For any graph $H$ and any coloring $c$ of its edges with two colors, there exists $n$ such that every $2$-coloring of the edges of the complete graph $K_n$ contains $H$ with every edge colored according to $c$ or none of its edges colored according to $c$.
I have been trying to draw a diagram to make sense of the question but am unable to do so and to proceed.