You are asking for a block design with universe size v=2n, block size k=n, any pair of elements in exactly lambda=n-1 blocks. (The latter demand seems pretty stupid, by the way, you should have allowed any lambda>0 and preferred smaller lambda.) Just look in standard tables of block design parameter sets. Here's one (out of date): http://www.uow.edu.au/~jennie/WEBPDF/032_1973.pdf The only slightly funny thing in your problem is that you demand the complement of a block must also be a block.
If we allow lambda to be arbitrary, then ANY n works: just list all the (2n-1)-choose-n ways to divide the 2n players into two teams. But I agree with Warren's basic point, that it might make sense to look at solutions with lambda bigger than 1. Jim Propp On Monday, November 16, 2015, Warren D Smith <warren.wds@gmail.com> wrote:
You are asking for a block design with universe size v=2n, block size k=n, any pair of elements in exactly lambda=n-1 blocks. (The latter demand seems pretty stupid, by the way, you should have allowed any lambda>0 and preferred smaller lambda.)
Just look in standard tables of block design parameter sets. Here's one (out of date): http://www.uow.edu.au/~jennie/WEBPDF/032_1973.pdf
The only slightly funny thing in your problem is that you demand the complement of a block must also be a block.
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