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Moreover, we prove that for $n$ in a set of density $1$, clique number is actually concentrated on a single value. As a simple consequence of these results, we also prove the one-point concentration result for the chromatic number, thus proving the $\\mathbb{F}_2^n$ analogue of the famous conjecture by Bollob\\'{a}s and giving alm"},"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":"1510.05991","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-10-20T18:08:26Z","cross_cats_sorted":[],"title_canon_sha256":"cca42146c19e44a61761155f1f4b9f847d8dff1af260eb7cf31e804dc618d126","abstract_canon_sha256":"04300958b0fab670322a313db534ea4c9b2e0e4662270dd05bb0b437e7a808a4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:29:39.191805Z","signature_b64":"EaTGzLsICjsV2PG56Jmw8NJ934KzV/UbD3LhNhcUhghSgQFJzXMvNv9N+NJH6R1/VpZoYEPUHN/aK8w6MktOCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"33e2a3f0ca437656def034e95a756a75d590393a0db8957195ddabda10a987c8","last_reissued_at":"2026-05-18T01:29:39.191065Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:29:39.191065Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"One-point concentration of the clique and chromatic numbers of the random Cayley graph on F_2^n","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Rudi Mrazovi\\'c","submitted_at":"2015-10-20T18:08:26Z","abstract_excerpt":"Green showed that there exist constants $C_1,C_2>0$ such that the clique number $\\omega$ of the random Cayley graph on $\\mathbb{F}_2^n$ satisfies $\\lim_{n\\to\\infty}\\mathbb{P}(C_1n\\log n < \\omega < C_2n\\log n)=1$. In this paper we find the best possible $C_1$ and $C_2$. Moreover, we prove that for $n$ in a set of density $1$, clique number is actually concentrated on a single value. As a simple consequence of these results, we also prove the one-point concentration result for the chromatic number, thus proving the $\\mathbb{F}_2^n$ analogue of the famous conjecture by Bollob\\'{a}s and giving alm"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.05991","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1510.05991","created_at":"2026-05-18T01:29:39.191178+00:00"},{"alias_kind":"arxiv_version","alias_value":"1510.05991v1","created_at":"2026-05-18T01:29:39.191178+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1510.05991","created_at":"2026-05-18T01:29:39.191178+00:00"},{"alias_kind":"pith_short_12","alias_value":"GPRKH4GKIN3F","created_at":"2026-05-18T12:29:22.688609+00:00"},{"alias_kind":"pith_short_16","alias_value":"GPRKH4GKIN3FNXXQ","created_at":"2026-05-18T12:29:22.688609+00:00"},{"alias_kind":"pith_short_8","alias_value":"GPRKH4GK","created_at":"2026-05-18T12:29:22.688609+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/GPRKH4GKIN3FNXXQGTUVU5LKOX","json":"https://pith.science/pith/GPRKH4GKIN3FNXXQGTUVU5LKOX.json","graph_json":"https://pith.science/api/pith-number/GPRKH4GKIN3FNXXQGTUVU5LKOX/graph.json","events_json":"https://pith.science/api/pith-number/GPRKH4GKIN3FNXXQGTUVU5LKOX/events.json","paper":"https://pith.science/paper/GPRKH4GK"},"agent_actions":{"view_html":"https://pith.science/pith/GPRKH4GKIN3FNXXQGTUVU5LKOX","download_json":"https://pith.science/pith/GPRKH4GKIN3FNXXQGTUVU5LKOX.json","view_paper":"https://pith.science/paper/GPRKH4GK","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1510.05991&json=true","fetch_graph":"https://pith.science/api/pith-number/GPRKH4GKIN3FNXXQGTUVU5LKOX/graph.json","fetch_events":"https://pith.science/api/pith-number/GPRKH4GKIN3FNXXQGTUVU5LKOX/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/GPRKH4GKIN3FNXXQGTUVU5LKOX/action/timestamp_anchor","attest_storage":"https://pith.science/pith/GPRKH4GKIN3FNXXQGTUVU5LKOX/action/storage_attestation","attest_author":"https://pith.science/pith/GPRKH4GKIN3FNXXQGTUVU5LKOX/action/author_attestation","sign_citation":"https://pith.science/pith/GPRKH4GKIN3FNXXQGTUVU5LKOX/action/citation_signature","submit_replication":"https://pith.science/pith/GPRKH4GKIN3FNXXQGTUVU5LKOX/action/replication_record"}},"created_at":"2026-05-18T01:29:39.191178+00:00","updated_at":"2026-05-18T01:29:39.191178+00:00"}