Kalai's 3^d conjecture

Kalai's 3-dimensional conjecture, proposed by Gil Kalai, pertains to the geometry of convex polytopes. The conjecture specifically addresses the conditions under which a simplicial complex can be realized as the nerve of a covering by open sets in a topological space. More concretely, it asserts that any simplicial complex that has a specific homotopy type will have a realization in a three-dimensional space.