Joint diamonds and Laver diamonds

TL;DR

This paper introduces the concept of joint guessing principles, particularly joint Laver diamonds, and explores their consistency and separation results across various large cardinal assumptions.

Contribution

It defines joint Laver diamonds, establishes their consistency with large cardinals, and demonstrates separations between different lengths of joint Laver diamonds.

Findings

Joint Laver diamonds are nontrivial and distinct from standard Laver diamonds.

Equiconsistency results are established for various large cardinals.

Sharp separations are proved between joint Laver diamonds of different lengths.

Abstract

The concept of jointness for guessing principles, specifically $\diamondsuit_\kappa$ and various Laver diamonds, is introduced. A family of guessing sequences is joint if the elements of any given sequence of targets may be simultaneously guessed by the members of the family. While equivalent in the case of $\diamondsuit_\kappa$, joint Laver diamonds are nontrivial new objects. We give equiconsistency results for most of the large cardinals under consideration and prove sharp separations between joint Laver diamonds of different lengths in the case of $\theta$-supercompact cardinals.