The word problem and the Aharoni-Berger-Ziv conjecture on the connectivity of independence complexes

For each finite simple graph $G$, Aharoni, Berger and Ziv consider a recursively defined number $\psi (G)\in \mathbb{Z}\cup \{+\mathrm{\infty }\}$ which gives a lower bound for the topological connectivity of the independence complex $I_G$. They conjecture that this bound is optimal for every graph. We use a result of recursion theory to give a short disproof of this claim.