Generalised Mycielski graphs, signature systems, and bounds on chromatic numbers

We prove that the coindex of the box complex B(H) of a graph H can be measured by the generalised Mycielski graphs which admit a homomorphism to H. As a consequence, we exhibit for every graph H a system of linear equations, solvable in polynomial time, with the following properties: if the system has no solutions, then coind(B(H))+2≤3; if the system has solutions, then χ(H)≥4. We generalise the method to other bounds on chromatic numbers using linear algebra.