Prikry-type forcings after collapsing a huge cardinal

Abstract

Some models of combinatorial principles have been obtained by collapsing a huge cardinal in the case of the successors of regular cardinals. For example, saturated ideals, Chang's conjecture, polarized partition relations, and transfer principles for chromatic numbers of graphs. In this paper, we study these in the case of the successors of singular cardinals. In particular, we show that Prikry forcing preserves the centeredness of ideals but kills the layeredness. We also study $\binom{\mu^{++}}{\mu^{+}}\to\binom{\kappa}{\mu^{+}}_{\mu}$ and $\mathrm{Tr}_{\mathrm{Chr}}(\mu^{+++},\mu^{+})$ in the extension by Prikry forcing at $\mu$.