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?