[math-fun] paintball

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 Here's one (out of date): http://www.uow.edu.au/~jennie/WEBPDF/032_1973.pdf The only slightly funny thing in your problem is that you demand the complement of a block must also be a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.