A “geometrical” representation for Ramsey's theorem

Question

The [infinite] Ramsey theorem states that

Let $n$ and $k$ be natural numbers. Every partition $\{X_1,\ldots ,X_k\}$ of $[\omega]^n$ into $k$ pieces has an infinite homogeneous set. Equivalently, for every $F\colon [\omega]^n\to \{1, . . . , k\}$ there exists an infinite $H \subseteq \omega$ such that $F$ is constant on $[H]^n$.

Where $[X]^k := \{ Y \subset X | |Y| = k\}$.

Now, when $k=2$ we can interpret this as coloring edges of a complete graph. But what happens when $k>2$, is there some geometrical or graphic, in a similar sense, to which we can turn this state into?

Answer 1

Partitioning $[\omega]^{(n)}$ into $k$ pieces can be interpreted as colouring the complete $n$-uniform hypergraph, with vertex set $\omega$, in $k$ colours.

When $n=2$ the complete $2$-uniform hypergraph is the complete graph with vertex set $\omega$.

For a definition of hypergraph see http://en.wikipedia.org/wiki/Hypergraph. Briefly it is the generalisation of a graph where edges can join more than two vertices. And by complete $n$-uniform I mean all the edges contain exactly $n$ vertices and every possible edge of size $n$ is in the hypergraph.