HIGHER-DIMENSIONAL HIGHLY CONNECTED RAMSEY THEORY (Set Theory : Reals and Topology)

Abstract

Bergfalk, Hrusak, and Shelah recently introduced a weakening of the classical partition relation for pairs in which the complete monochromatic subgraph of the classical relation is replaced by a highly connected monochromatic subgraph. In subsequent work, we proved that, assuming the consistency of the existence of a weakly compact cardinal, it is consistent that an optimal square-bracket version of this highly connected partition relation holds at the continuum. In this paper, we introduce a higher-dimensional generalization of the highly connected partition relation and prove an analogous consistency result indicating that, if the existence of a weakly compact cardinal is consistent, then it is consistent that an optimal square-bracket version of the higherdimensional highly connected partition relation holds at the continuum.