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For a finite simple graph with at least one vertex, the complement theta number has the equivalent semidefinite program formulationsandThe Schur complement and $U=tZ-J$ prove the equivalence; any feasible $t$ is at least one, so division by $t$ is valid. For a graph with an edge, $U/(t-1)$ is the Gram matrix of a strict vector coloring. Consequently $\bar{\vartheta }(G)$ is the strict vector chromatic number, and an ordinary $k$-graph colouring gives $\bar{\vartheta }(G)\le k$ by assigning the colors the vertices of a regular simplex.

For a proper three-graph colouring $c$, associate the colors with the three roots of unity $d(i)=\exp (2\pi i(c(i)-1)/3)$, viewed as unit vectors in ${\mathbb{R}}^2$. Define $U_{ij}=2⟨d(i),d(j)⟩$ using the real inner product.

This is twice a Gram matrix, so it is a positive semidefinite matrix. Its diagonal is $2$. On every edge, the endpoint colors differ, so their vectors make angle $2\pi /3$ or $4\pi /3$ and $U_{ij}=2\cos (2\pi /3)=-1$. Thus $t=3$ and this $U$ are feasible in the second semidefinite program, provingThis is the regular simplex construction of a strict vector coloring.