Failure of an higher analogue of Mho

TL;DR

This paper demonstrates that a stronger analogue of Moore's weak club-guessing principle at the second uncountable cardinal can consistently fail, using advanced forcing techniques starting from large cardinals.

Contribution

It introduces a novel forcing construction showing the possible failure of the stronger analogue of $ ho$ at the second uncountable cardinal.

Findings

The stronger analogue of $ ho$ can fail consistently.

Forcing construction uses supercompact and inaccessible cardinals.

Violates the analogue at the second uncountable cardinal.

Abstract

Justin Moore's weak club-guessing principle $\mho$ admits various possible generalizations to the second uncountable cardinal. One of them was shown to hold in ZFC by Shelah. A stronger one was shown to follow from several consequences of the continuum hypothesis by Inamdar and Rinot. Here we prove that the stronger one may consistently fail. Specifically, starting with a supercompact cardinal and an inaccessible cardinal above it, we devise a notion of forcing consisting of finite working parts and finitely many two types of models as side conditions, to violate this analog of $\mho$ at the second uncountable cardinal.