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jackHaving failed to use the biggest hammer ("ask the internet if anyone can think of a general mathematical solution"), I tried the next-biggest, brute-force.
I wrote a program which divides people into random matchings, and then switches matches which are over- or -under represented. I'm not quite sure what sort of "random switching" is best, I hoped to just get lucky.
The first effort found a version for 9 people easily, but didn't find one any larger than that. The second was about the same. That's where I am now. I can describe the shuffling in detail if anyone is interested or thinks they might have any suggestions.
I wrote a program which divides people into random matchings, and then switches matches which are over- or -under represented. I'm not quite sure what sort of "random switching" is best, I hoped to just get lucky.
The first effort found a version for 9 people easily, but didn't find one any larger than that. The second was about the same. That's where I am now. I can describe the shuffling in detail if anyone is interested or thinks they might have any suggestions.