# What kind of problem is picking sets of sets of things that don't overlap?

I was working out some combinations for a game, when I hit upon a problem that seems unsolvable. I'm wondering if some existing mathematics (probably in graph theory or combinatorics) covers this particular class of problem. To express it without any game-specific terminology:

I have a set of 6 numbers {1,2,3,4,5,6} from which I will create 3 number combinations. These combinations are always ordered and never contain duplicates, such as {123,134,145,156...}. Simple combinations tells me there are 20 such configurations.

Now from those configurations, I wish to select sets of 4 each, such that no two numbers recur in any of the 4 triplets. An example would be {123,145,246,235}.

My goal was to sort all 20 combinations into 5 sets of 4, but I don't think it's possible. I always end up with 2 pairs that repeat, either in 2 separate sets or both in one set, like {123,145,236,245} (23 and 45 occur twice). Note that being split still counts, so {234,245} is still a failure because 24 repeats.

I figure this problem must be analogous to a node walking problem or something like that, but I don't know the math well enough to prove that no solution exists or to find similar problems by searching.

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