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In this note we resolve their problem by showing that $I_{K_{2,2,1}}(e)$ has at least two local maxima in $(0,1)$. Additionally, we determine $I_{K_{2,2,1}}(e)$, when $e=(k-1)/k$ for every integer $k\\ge 3.$"},"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":"2605.15021","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-05-14T16:21:33Z","cross_cats_sorted":[],"title_canon_sha256":"e2f9e0136840ab20d88dd9334d91ee42ed39ebfca252baddfa310ff320de4dc4","abstract_canon_sha256":"1f94f2b86d8da14640655f64e18825e4ac04e0badbf0f06d426bbc715171368b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:38:54.681818Z","signature_b64":"0lchdOUYHn+q4vRg7Ah3gNeTNWXDPimd59ZLTzjyopJUKjLtmIFE9IlDavOMvteh7L+eCXFEMsOd+eRuFZ85Dg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3006fbbdf311c0ade03f2f666c3ed5dd0519804d2bc27a10fc8de4cfe90381ac","last_reissued_at":"2026-05-17T23:38:54.681119Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:38:54.681119Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Local maximum of inducibility profiles","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Bernard Lidick\\'y, J\\'ozsef Balogh","submitted_at":"2026-05-14T16:21:33Z","abstract_excerpt":"For a graph $G$ and $e\\in [0,1]$, denote by $I_G(e)$ the supremum of densities of $G$ over $n$-vertex graphs with edge density $e$ as $n$ goes to infinity. 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