We study the number of normal measures on a tall cardinal. Our main results are that:

• • The least tall cardinal may coincide with the least measurable cardinal and carry as many normal measures as desired.

• • The least measurable limit of tall cardinals may carry as many normal measures as desired.

Three central combinatorial properties in set theory are the tree property, the approachability property and stationary reflection. We prove the mutual independence of these properties by showing that any of their eight Boolean combinations can be forced to hold at ${\kappa ^{ + + }}$, assuming that $\kappa = {\kappa ^{ < \kappa }}$ and there is a weakly compact cardinal above κ.

If in addition κ is supercompact then we can force κ to be ${\aleph _\omega }$ in the extension. The proofs combine the techniques of adding and then destroying a nonreflecting stationary set or a ${\kappa ^{ + + }}$-Souslin tree, variants of Mitchell’s forcing to obtain the tree property, together with the Prikry-collapse poset for turning a large cardinal into ${\aleph _\omega }$.

From large cardinals we show the consistency of normal, fine, κ-complete λ-dense ideals on ${{\cal P}_\kappa }\left( \lambda \right)$ for successor κ. We explore the interplay between dense ideals, cardinal arithmetic, and squares, answering some open questions of Foreman.