Forcing and Construction Schemes

We investigate forcing and independence questions relating to construction schemes. We show that adding $\kappa \ge {\omega }_1$ 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 $n\le m$ and incompatible if $n>m$.