In my book about graph theory it quite casually states that for the smallest dimension $d$ for which there exists an orthonormal representation of a graph $G$ it holds that $d \leq \chi(\bar{G})$ (where $\chi(\bar{G})$ is the coloring of the graph $\bar{G}$.
An orthonormal representation of a graph $G$ is a set of unit vectors $\{u_1, \dots u_n\} \subseteq \mathbb{R}^d $ with $u_i^Tu_j = 0$ when $\{i,j\}\in \bar{E}$.
How does this work?
Say $S\subset V$ is the maximum stable set of $G$. When we can form an orthogonal representation in the following way: for all $i,j\in S$ we need $u_i^Tu_j =0$ because $\{i,j\}\in \bar{E}$. Say $S = \{v_1, \dots v_{|S|}\}$, set $u_{v_k} = e_k$ (so the canononical base vector with all zeros except on the $k^{th}$ component.
Then this is an orthagonal representation of $G$, but $$\min\{d |\{u_1, \dots u_n\}\in \mathbb{R}^d \text{ orthogonal representation }G \} \leq |S| =\alpha(G) = \omega(\bar{G}) \leq \chi(\bar{G})$$
and therfore smallest dimension for which there exists an orthogonal representation bounded by the coloring of the complement.