[bazzargh]

it works for any number of integers. The first proof above (before tl;dr) is showing that every positive integer has a unique representation as a sum of distinct powers of 2, ie binary, and that no two integers have the same representation. You can watch a lecture about the representation of sets in binary here https://www.youtube.com/watch?v=Iw21xgyN9To (google representing sets with bits for way more like this)

But again it's not useful in practice for very sparse sets: if you have say a million players, with at most 10 at the same poker table, setting 10 bits of a million-bit binary number is super wasteful. Even representing the players as fixed size 20-bit numbers (1 million in binary is 20 bits long), and appending the 10 sorted numbers, means you don't need more than 200 bits to represent this set.

And you can go much smaller if all you want is to label a _bucket_ that includes this particular set; just hash the 10 numbers to get a short id. Then to query faster for a specific combination of players you construct the hash of that group, query to get everything in that bucket (which may include false positives), then filter this much smaller set of answers.