IMO Shortlist 2014 problem C6
We are given an infinite deck of cards, each with a real number on it. For every real number , there is exacly one card in the deck that has written on it. Now two players draw disjoint sets and of cards each from this deck. We would like to define a rule that declares one of them a winner. This rule should satisfy the following conditions:
How many ways are there define such a rule? Here, we consider the rules as different if there exist two sets and such that beats according to the rule, but beats according to the other.