In extension to this question 
https://mathoverflow.net/questions/86550/positive-but-not-completely-positive
I'd like to know, for $k>1$, examples of $k$-positive linear maps of a matrix algebra into itself that are not $k+1$-positive. (I don't know a single one.) By [M.D. Choi's theorem](https://en.wikipedia.org/wiki/Choi's_theorem_on_completely_positive_maps)
the size of the matrices involved must grow with $k$, but how fast?