TITLE: Does minimal degree $n$ imply a $K_n$ minor
QUESTION [10 upvotes]: Is it true that any finite graph has a $K_n$ minor, where $n$ is a minimal vertex degree?

REPLY [22 votes]: More generally, it is a classic result (independently due to Kostochka and Thomason) that minimum degree $(\alpha+o(1))n \sqrt{\log n}$ suffices to force a $K_n$ minor, where $\alpha$ is an explicit constant.  Conversely, there are random graphs with minimum degree $\Omega(n\sqrt{\log n})$ that do not contain a $K_n$ minor.   See here to access the paper by Thomason.
Update. Alon, Krivelevich, and Sudakov have recently given a new short proof of the Kostochka-Thomason result.