Khot, Subhash; Minzer, Dor; Safra, Muli Pseudorandom sets in Grassmann graph have near-perfect expansion. (English) Zbl 07690461 Ann. Math. (2) 198, No. 1, 1-92 (2023). Summary: 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 \(\mathsf{Gr}_{\mathsf{global}}\) contains induced subgraphs \(\mathsf{Gr}_{\mathsf{local}}\) that are themselves isomorphic to Grassmann graphs of lower orders. A set is called pseudorandom if its density is \(o(1)\) inside all subgraphs \(\mathsf{Gr}_{\mathsf{local}}\) whose order is \(O(1)\) lower than that of \(\mathsf{Gr}_{\mathsf{global}}\). 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. Cited in 16 Documents MSC: 68-XX Computer science Keywords:probabilistically checkable proofs; unique-games conjecture; small-set expansion; hypercontractivity × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Arora, Sanjeev; Lund, Carsten, Approximation Algorithms for {NP-hard} Problems (1996) · Zbl 1368.68010 · doi:10.1145/261342.571216 [2] Arora, Sanjeev; Lund, Carsten; Motwani, Rajeev; Sudan, Madhu; Szegedy, Mario, Proof verification and the hardness of approximation problems, J. ACM. Journal of the ACM, 45, 501-555 (1998) · Zbl 1065.68570 · doi:10.1145/278298.278306 [3] Arora, Sanjeev; Safra, Shmuel, Probabilistic checking of proofs: a new characterization of {NP}, J. ACM. 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