@ZackWolske - thanks, that is helpful. If I found the Warwick Harvey link that you are referring to, then the actual website is down (at least for now). The computer program looks promising, though I am not very good with tech and don't know how to implement it. If somebody did, I'd be happy to accept that answer, although I would be interested in conceptual discussion - original question edited to reflect that. (As hinted above, I really am just echoing a question I received from a friend not in math so ... I am not a researcher in this topic.)

@The Masked Avenger - I apologize for the lack of clarity. I anticipated that when I asked "What is the maximal number of compatible partitions you can form," it would be interpreted literally, instead of as "What is a trivial upper bound on the maximal number of compatible partitions you can form." Anyhow, let me clarify now that I am indeed asking for the actual answer to the actual question. That being said, I would certainly be interested in nontrivial upper and lower bounds.

@Zack - thanks for the link. It seems interesting, but after glancing through the first one I don't see an answer to the instance of my question, or a way to get compute it. As you yourself point out, the problem is slightly different - the SGP asks for a perfect'' solution, and I am asking for an optimal'' solution. I am unable to see the published paper for subscription reasons. If you have read it, do you know if it addresses methods for my problem?