Homotopy types of box complexes

In [MZ04] Matousek and Ziegler compared various topological lower bounds for the chromatic number. They proved that Lovasz's original bound [L78] can be restated as $\chr G \geq \ind (\B(G)) +2$. Sarkaria's bound [S90] can be formulated as $\chr G \geq \ind (\B_0(G)) +1$. It is known that these lower bounds are close to each other, namely the difference between them is at most 1. In this paper we study these lower bounds, and the homotopy types of box complexes. Some of the results was announced in [MZ04].