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We prove that the coindex of the box complex $\mathrm{B}(H)$ of a graph $H$can be measured by the generalised Mycielski graphs which admit a homomorphismto it. As a consequence, we exhibit for every graph $H$ a system of linearequations solvable in polynomial time, with the following properties: If thesystem has no solutions, then $\mathrm{coind}(\mathrm{B}(H)) + 2 \leq 3$; ifthe system has solutions, then $\chi(H) \geq 4$. We generalise the method toother bounds on chromatic numbers using linear algebra.

Generalised Mycielski graphs and bounds on chromatic numbers