We prove that pseudorandom sets in Grassmann graph have near-perfect expansion as hypothesized in [DKKMS-2]. This completes the proof of the 2 -to- 2 Games Conjecture (albeit with imperfect completeness) as proposed in [KMS, DKKMS-1], along with a contribution from [BKT].

The Grassmann graph G r globa l contains induced subgraphs G r loca l that are themselves isomorphic to Grassmann graphs of lower orders. A set is called pseudorandom if its density is o (1) inside all subgraphs G r loca l whose order is O (1) lower than that of G r globa l . We prove that pseudorandom sets have expansion 1 − o (1) , greatly extending the results and techniques in [DKKMS-2].