Pseudorandom Sets in Grassmann Graph have Near-Perfect Expansion

We prove that pseudorandom sets in Grassmann graph have near-perfect expansion as hypothesized in [4]. This completes the proof of the 2-to-2 Games Conjecture (albeit with imperfect completeness) as proposed in [15, 3], along with a contribution from [2, 14]. 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 in [4]. ∗Department of Computer Science, Courant Institute of Mathematical Sciences, New York University. Research supported by NSF CCF-1422159, Simons Collaboration on Algorithms and Geometry, and Simons Investigator Award. †School of Computer Science, Tel Aviv University. Supported by Clore Fellowship. ‡School of Computer Science, Tel Aviv University. ISSN 1433-8092 Electronic Colloquium on Computational Complexity, Revision 1 of Report No. 6 (2018)