Pseudorandom sets in Grassmann graph have near-perfect expansion

We prove that pseudorandom sets in Grassmann graph have near-perfect expansion. This completes the proof of the 2-to-2 Games Conjecture (albeit with imperfect completeness). Some implications of this new result are improved hardness results for Minimum Vertex Cover, improving on the work of Dinur and Safra [Ann. of Math. 162 (2005), 439-485], and new hardness gaps for Unique-Games. The Grassmann graph Grglobal contains induced subgraphs Grlocal that are themselves isomorphic to Grassmann graphs of lower orders. A set is called pseudorandom if its density is o(1) inside all subgraphs Grlocal whose order is O(1) lower than that of Grglobal. We prove that pseudorandom sets have expansion 1 - o(1), greatly extending the results and techniques of a previous work of the authors with Dinur and Kindler.