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We show that for all $F$ on at least three vertices and all $p \\in (0,1)$, the binomial random graph $G(n,p)$ has induced $F$-density strictly less than $I(F,p).$ This provides a negative answer to a problem posed by Liu, Mubayi and Reiher.\n  Our approach is in the limiting setting of graphons, and we in fact show a stronger result: the binomial random graph is never a \\emph{"},"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":"2306.13014","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2023-06-22T16:23:21Z","cross_cats_sorted":[],"title_canon_sha256":"289f2ed0a8cfe21cd0dc1bf7a470bd54a18219da090c7d4d84e195b78f9f2bc2","abstract_canon_sha256":"dbf45698ac7b5688847f6eca925ed98b7c43745a110f030accfc4f40ca83c071"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T08:58:19.681378Z","signature_b64":"MWW0L0w4chuzVH77v+zPRopWD8Ey12i3K/eaT/XPOtFVJhgEn2Dy4tDjA5ml4yXCBlZEPbcm3MJnij6b2OcXAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ec8196f41368e4261b3a0591422ba3eb3e7b9cfd195f744caec5f556ff6e706d","last_reissued_at":"2026-07-05T08:58:19.680895Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T08:58:19.680895Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The binomial random graph is a bad inducer","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Fan Wei, Marcus Michelen, Vishesh Jain","submitted_at":"2023-06-22T16:23:21Z","abstract_excerpt":"For a finite graph $F$ and a value $p \\in [0,1]$, let $I(F,p)$ denote the largest $y$ for which there is a sequence of graphs of edge density approaching $p$ so that the induced $F$-density of the sequence approaches $y$. 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