Forcing and Construction Schemes

TL;DR

This paper explores the relationships between forcing, construction schemes, and combinatorial principles, showing how adding Cohen reals influences the existence of capturing schemes and their hierarchy.

Contribution

It demonstrates the consistency of having n-capturing schemes without (n+1)-capturing schemes and clarifies their relation to the m-Knaster hierarchy.

Findings

Adding Cohen reals creates capturing construction schemes.

n-capturing schemes can exist without (n+1)-capturing schemes.

Independence and incompatibility results between n-capturing and m-Knaster principles.

Abstract

We investigate forcing and independence questions relating to construction schemes. We show that adding $\kappa\geq\omega_1$ Cohen reals adds a capturing construction scheme. We study the weaker structure of $n$-capturing construction schemes and show that it is consistent to have $n$-capturing construction schemes but no $(n+1)$-capturing construction schemes. We also study the relation of $n$-capturing with the $m$-Knaster hierarchy and show that MA$_{\omega_1}($K$_m)$ and $n$-capturing are independent if $n\leq m$ and incompatible if $n>m$.