TITLE: Can we have a $\kappa$-Suslin tree where $\kappa$ is above a measurable cardinal?
QUESTION [6 upvotes]: Question: Can we have a set theory in which there exists a $\kappa$-Suslin tree with $\kappa$ larger than the least measurable cardinal?
A $\kappa$-Suslin tree is a tree with levels indexed by $\kappa$, the cardinality of each level is less than $\kappa$, and all chains and antichains have cardinality less than $\kappa$. Essentially, the only thing I could find is the Jensen theorem that if $V=L$ then there exists a $\kappa$-Suslin tree for every infinite successor cardinal $\kappa$. But $V=L$ excludes measurable cardinals.

REPLY [8 votes]: To complement Joel's answer and tie into the issue of $L$, consider a model $L[U]$, where $U$ is a normal measure on a cardinal $\kappa$.  It is well-known that $L[U]$ satisfies GCH and $\diamondsuit_\mu(S)$ for every regular $\mu$ and every stationary $S \subseteq \mu$.  These principles suffice to construct $\mu$-Suslin trees whenever $\mu$ is a successor of a regular cardinal.