Printed Carton Quality

I doubt very much that you will find a standard that tells you what PPM level will be acceptable for you.  Only you can decide that.  If you need measurement techniques, I would look at ASTM, TAPPI, and similar organizations.  In the end, if you are having problems with a supplier you need to either get them to grow or go.  This may mean your quality organization needs to make a more active role in dealing with these issues by actually going to the supplier and identifying what the underlying root causes are. 

... on the other hand, if the supplier was worth their salt they'd have fixed it by now.