Connector-free lemma (Bapat)

Statement

A connector of a simplex is a connected set that touches all n faces of the simplex.

A connector-free covering is a covering $C_1,\ldots,C_n$ in which no $C_i$ contains a connector.

Any KKM covering is a connector-free covering, since in a KKM covering, no $C_i$ even touches all n faces. However, there are connector-free coverings that are not KKM coverings. An example is illustrated at the right. There, the red set touches all three faces, but it does not contain any connector, since no connected component of it touches all three faces.

A theorem of Ravindra Bapat, generalizing Sperner's lemma, implies the KKM lemma extends to connector-free coverings (he proved his theorem for $n=3$).

The connector-free variant also has a permutation variant, so that both these generalizations can be used simultaneously.