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. ${\bf 162}$ (2005), 439--485], and new hardness gaps for Unique-Games. The Grassmann graph ${\sf Gr}_{\sf{global}}$ contains induced subgraphs ${\sf Gr}_{\sf{local}}$ that are themselves isomorphic to Grassmann graphs of lower orders. A set is called pseudorandom if its density is $o(1)$ inside all subgraphs ${\sf Gr}_{\sf{local}}$ whose order is $O(1)$ lower than that of ${\sf Gr}_{\sf{global}}$. 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.