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

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