A Generalization of Gale's lemma

Abstract

In this work, we present a generalization of Gale's lemma. Using this generalization, we introduce two combinatorial sharp lower bounds for $\mathrm{conid}({\mathrm{B}}_0(G))+1$ and $\mathrm{conid}(\mathrm{B}(G))+2$, two famous topological lower bounds for the chromatic number of a graph $G$.