Some combinatorial properties of splitting trees

TL;DR

This paper investigates the combinatorial properties of splitting trees, demonstrating the absence of the weak Sacks property in splitting forcing, analyzing partition results, and determining cardinal invariants after specific forcing iterations.

Contribution

It provides new insights into the behavior of splitting forcing, including the absence of the weak Sacks property and the effects on cardinal invariants after iterations.

Findings

Splitting forcing lacks the weak Sacks property below any condition.

Partition results for splitting trees can hold or fail depending on circumstances.

Cardinal invariants are determined after an $oldsymbol{ extomega_2}$-length countable support iteration.

Abstract

We show that splitting forcing does not have the weak Sacks property below any condition, answering a question of Laguzzi, Mildenberger and Stuber-Rousselle. We also show how some partition results for splitting trees hold or fail and we determine the value of cardinal invariants after an $\omega_2$-length countable support iteration of splitting forcing.