Consider a binary relation $R$ over a finite set $X$ of size $n$. Assume $R$ is antisymmetric and connected but not necessarily transitive. In essence, we are modeling an "option x beats option y" relation, which is not necessarily transitive. It might be the result of a voting process for instance.

It is sometimes possible to find a function $g : X \rightarrow \mathbf{R}^d$ which assigns a $d$ dimensional real vector to each element of X, and a continuous function $f: \mathbb{R}^{n + n} \rightarrow \mathbb{R}$ monotonous in each of its coordinates, such that $xRy \Leftrightarrow f(g(x),g(y)) > 0$.

Taking $d = n$ and $g$ a one-hot encoding, it's easy to construct $f$.

This is not generally possible for all $d$. If $d = 1$, this is possible iff $R$ is transitive.

Is there a fixed $d$ such that, for all $n$ and all $R$, $g$ and $f$ always exist?