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More precisely, let $F$ be a graph with $m>0$ edges, and let $n$ be the number of non-isolated vertices of $F$. If $$\n  p\\ge \\binom {n}{2}/m, $$ then for every $\\rho$-locally dense graphon $W$, $$\n  t(F,W^{\\circ p})\\ge \\rho^{pm}. $$ Equivalently, if $$\n  W_F(\\"},"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":"2606.30010","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-06-29T09:16:50Z","cross_cats_sorted":[],"title_canon_sha256":"07dbc706c84c445f126be9ad317d0bb68b7773e94d836ab8aaa59cf6e0c88665","abstract_canon_sha256":"1bb3a24a749f4ec80bbd84d0392123b5b9487ea81e210f22120e5a5c6aa842ec"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-30T02:17:45.620948Z","signature_b64":"UAwwP5EMdtBYucPypmlEQGbYi+y/e5PzeboFZ91YRaJ0b1slUdSign8godgsfyCaUDCnNwzoQssaQbmI3EcDCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5b536d993d0d62fe270c11651fe534994545f69b42f5a2b4bae03f6258f1fc5e","last_reissued_at":"2026-06-30T02:17:45.620328Z","signature_status":"signed_v1","first_computed_at":"2026-06-30T02:17:45.620328Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"$L^p$-form of the KNRS conjecture","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Yuqi Zhao","submitted_at":"2026-06-29T09:16:50Z","abstract_excerpt":"The Kohayakawa--Nagle--R\\\"odl--Schacht conjecture predicts that locally dense graphs contain, asymptotically, at least as many homomorphic copies of any fixed graph as the random graph of the same edge density. We prove that every graph with at least one edge satisfies a natural $L^p$ relaxation of this conjecture in the graphon setting. More precisely, let $F$ be a graph with $m>0$ edges, and let $n$ be the number of non-isolated vertices of $F$. If $$\n  p\\ge \\binom {n}{2}/m, $$ then for every $\\rho$-locally dense graphon $W$, $$\n  t(F,W^{\\circ p})\\ge \\rho^{pm}. $$ Equivalently, if $$\n  W_F(\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.30010","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.30010/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2606.30010","created_at":"2026-06-30T02:17:45.620424+00:00"},{"alias_kind":"arxiv_version","alias_value":"2606.30010v1","created_at":"2026-06-30T02:17:45.620424+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.30010","created_at":"2026-06-30T02:17:45.620424+00:00"},{"alias_kind":"pith_short_12","alias_value":"LNJW3GJ5BVRP","created_at":"2026-06-30T02:17:45.620424+00:00"},{"alias_kind":"pith_short_16","alias_value":"LNJW3GJ5BVRP4JYM","created_at":"2026-06-30T02:17:45.620424+00:00"},{"alias_kind":"pith_short_8","alias_value":"LNJW3GJ5","created_at":"2026-06-30T02:17:45.620424+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2607.02260","citing_title":"Tensor Amplification and Spectral Transfer for Sidorenko-Type Inequalities","ref_index":20,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/LNJW3GJ5BVRP4JYMCFSR7ZJUTF","json":"https://pith.science/pith/LNJW3GJ5BVRP4JYMCFSR7ZJUTF.json","graph_json":"https://pith.science/api/pith-number/LNJW3GJ5BVRP4JYMCFSR7ZJUTF/graph.json","events_json":"https://pith.science/api/pith-number/LNJW3GJ5BVRP4JYMCFSR7ZJUTF/events.json","paper":"https://pith.science/paper/LNJW3GJ5"},"agent_actions":{"view_html":"https://pith.science/pith/LNJW3GJ5BVRP4JYMCFSR7ZJUTF","download_json":"https://pith.science/pith/LNJW3GJ5BVRP4JYMCFSR7ZJUTF.json","view_paper":"https://pith.science/paper/LNJW3GJ5","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2606.30010&json=true","fetch_graph":"https://pith.science/api/pith-number/LNJW3GJ5BVRP4JYMCFSR7ZJUTF/graph.json","fetch_events":"https://pith.science/api/pith-number/LNJW3GJ5BVRP4JYMCFSR7ZJUTF/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/LNJW3GJ5BVRP4JYMCFSR7ZJUTF/action/timestamp_anchor","attest_storage":"https://pith.science/pith/LNJW3GJ5BVRP4JYMCFSR7ZJUTF/action/storage_attestation","attest_author":"https://pith.science/pith/LNJW3GJ5BVRP4JYMCFSR7ZJUTF/action/author_attestation","sign_citation":"https://pith.science/pith/LNJW3GJ5BVRP4JYMCFSR7ZJUTF/action/citation_signature","submit_replication":"https://pith.science/pith/LNJW3GJ5BVRP4JYMCFSR7ZJUTF/action/replication_record"}},"created_at":"2026-06-30T02:17:45.620424+00:00","updated_at":"2026-06-30T02:17:45.620424+00:00"}