On Perles' question

The principal aim of this paper is to determine the minimal dimension that a nerve of the cover of the sphere S^h by the open sets not containing a pair of antipodal points could have, and also to determine the minimal cardinality of such cover (or the minimal number of vertices of its nerve). In particular, our result provides the complete answer to the question posed by Micha Perles. Our results could be seen as the extensions of the Lyusternik-Schnirel’man version of the Borsuk-Ulam theorem. As a consequence, we also obtain the improved lower bound for the local chromatic number of certain class of graphs.