# How do I apply the infinite Ramsey theorem to graph theory?

The infinite Ramsey theorem reads as such:

Theorem. Let $$X$$ be some infinite set and colour the elements of $$X^{(n)}$$ (the subsets of $$X$$ of size $$n$$) in $$c$$ different colours. Then there exists some infinite subset $$M$$ of $$X$$ such that the size $$n$$ subsets of $$M$$ all have the same colour.

If I take $$n=2$$, I can identify $$X$$ with the vertices of a simple graph and $$X^{(2)}$$ as the possible edges. If I now take $$c=2$$, I can intepret the result as: "Any graph on infinitely many vertices has an infinite clique or an infinite independent set."

But what would the interpretations be for $$c > 2$$? There are more applications to the infinite Ramsey theorem in extremal graph theory, right?

• Does coloring the edges of a graph not count as an interpretation? May 30, 2019 at 16:47
• @MishaLavrov "There is always a colour used infinitely often in any colouring of $K_\omega$" is a result I can deduce in two sentences without using the theorem. Do you have something more particular in mind?
– SK19
May 30, 2019 at 17:42

The standard interpretation of Ramsey's theorem for $$c>2$$ is that whenever the edges of $$K_\omega$$ are colored by finitely many colors, there is a monochromatic infinite clique in one of the colors.
(This is how we interpret $$c>2$$ in the finite case, too.)
Where the colors come from in any particular application of Ramsey's theorem is a different matter. "Colors" are just a more intuitive metaphor for partitioning the edge set of $$K_\omega$$, or for a function from $$E(K_\omega)$$ to a finite set.