Consider the oriented n-dimensional hypercubes $C_n$.

- $C_0$: one object $X_0$.
- $C_1$: $X_0 \to X_1$.
- $C_2$:

$$ \begin{array}{ccc} X_{00} & \rightarrow & X_{01} \\ \downarrow && \downarrow \\ X_{10} & \rightarrow & X_{11}. \end{array} $$ - $C_3$:

$$ \begin{array}{ccccccc} X_{000} & \rightarrow & \rightarrow & \rightarrow & X_{010} && \\ \downarrow & \searrow & & & \downarrow & \searrow & \\ \downarrow & & X_{100} & \rightarrow & \rightarrow &\rightarrow & X_{110} \\ \downarrow & & \downarrow & & \downarrow & & \downarrow \\ X_{001} & \rightarrow & \downarrow & \rightarrow & X_{011} && \downarrow \\ & \searrow & \downarrow & & & \searrow & \downarrow \\ & & X_{101} & \rightarrow & \rightarrow & \rightarrow & X_{111} \end{array} $$

And so on, inductively over $n \in \mathbb{N}$. Some of the objects (or everyone) filling the vertices can eventually be the same. Moreover, they can be $0$.

Let $I$ be an arbitrary finite oriented diagram (graph) with no cycles (composable arrows starting and ending on one single object).

My question is: Does always exist an $n \in \mathbb{N}$ such that $I$ is included in $C_n$? (References are also welcomed.)

Example 1: $I = \{ X \overset{f}{\underset{g}\rightrightarrows} Y \}$ can be
arranged on $C_2$:

$$ \begin{array}{ccc}
X & \overset{f}\rightarrow & Y \\
{\scriptstyle g}\downarrow && \downarrow \\
Y & \rightarrow & 0.
\end{array} $$

`\begin{center}`

is not valid here --- only math-style commands, because this is html, not real TeX --- let alone`\xymatrix`

. There is an effort to implement`\xymatrix`

into MathJax (the package that we will probably eventually switch to for TeX support), but it's not there yet. – Theo Johnson-Freyd Aug 9 '10 at 2:48