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Huang, Linial, Naves, Peled and Sudakov have recently determined the projection of the set S to the (d_0,d_3) plane. We determine the projection of the set S to all the remaining planes."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1610.02446","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-10-07T23:43:14Z","cross_cats_sorted":[],"title_canon_sha256":"36eb430f3d0cd3115fbaa1832e2c221f7b9d1caa135e896217f2fc1c77a47799","abstract_canon_sha256":"fcbeeb12071f6b38093e557939661b357fad72cd346105e5d653298e4e1b3ca9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:46:39.523716Z","signature_b64":"OszE2g6g3z7ovoW5qg7EfypWyMZsfeRxQgmaE80v8SbSXEbnTGHM8YWHfUEXUlKQTaVa96TX6r29j5o0dQW8BQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fbf6761149c0066ee5a185d01e223b04bb0b3b76d7166a430cdb95c98da22e01","last_reissued_at":"2026-05-18T00:46:39.523010Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:46:39.523010Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Densities of 3-vertex graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Andrzej Grzesik, Daniel Kral, Jan Volec, Ping Hu, Roman Glebov, Tamas Hubai","submitted_at":"2016-10-07T23:43:14Z","abstract_excerpt":"Let d_i(G) be the density of the 3-vertex i-edge graph in a graph G, i.e., the probability that three random vertices induce a subgraph with i edges. 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