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We show that the methods of Zhang can be further boosted, and prove the following strengthening. If $G$ is a graph with $m$ edges and no clique of size $m^{1/2-\\delta}$, then $G$ has a cut of size at least $m/2+m^{1/2+\\varepsilon}$ for some $\\varepsilon=\\varepsilon(\\delta)>0$.\n  In addition, we sharpen another result of Zhang by proving that if $G$ "},"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":"2507.13298","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2025-07-17T17:06:37Z","cross_cats_sorted":[],"title_canon_sha256":"4851128744d535f2dd53b755b46eb90040f40661eb6ec116165cdb3ae5acf297","abstract_canon_sha256":"e8007358481bdc981d827de669f31d9aac0fcdbe91a21fa2cc7bb0b08136fa3a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:38:58.674825Z","signature_b64":"WPbLXV9eGHfbHuVCnhNaw+upaMVpHbBwjyN5hZf3JVmWokNvM1lLQdxB09cYsPgdotStbxeoekX3ote9q5PUBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"157e194e96516867899a6fb1f654ccc5aa28cbe762866d8b5778719b21359937","last_reissued_at":"2026-07-05T11:38:58.674399Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:38:58.674399Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Beyond the MaxCut problem in $H$-free graphs","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Aleksa Milojevi\\'c, Istv\\'an Tomon, Zhihan Jin","submitted_at":"2025-07-17T17:06:37Z","abstract_excerpt":"In a recent breakthrough, Zhang proves that if $G$ is an $H$-free graph with $m$ edges, then $G$ has a cut of size at least $m/2+c_Hm^{0.5001}$, making a significant step towards a well known conjecture of Alon, Bollob\\'as, Krivelevich and Sudakov. We show that the methods of Zhang can be further boosted, and prove the following strengthening. If $G$ is a graph with $m$ edges and no clique of size $m^{1/2-\\delta}$, then $G$ has a cut of size at least $m/2+m^{1/2+\\varepsilon}$ for some $\\varepsilon=\\varepsilon(\\delta)>0$.\n  In addition, we sharpen another result of Zhang by proving that if $G$ "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.13298","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/2507.13298/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":"2507.13298","created_at":"2026-07-05T11:38:58.674445+00:00"},{"alias_kind":"arxiv_version","alias_value":"2507.13298v1","created_at":"2026-07-05T11:38:58.674445+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2507.13298","created_at":"2026-07-05T11:38:58.674445+00:00"},{"alias_kind":"pith_short_12","alias_value":"CV7BSTUWKFUG","created_at":"2026-07-05T11:38:58.674445+00:00"},{"alias_kind":"pith_short_16","alias_value":"CV7BSTUWKFUGPCM2","created_at":"2026-07-05T11:38:58.674445+00:00"},{"alias_kind":"pith_short_8","alias_value":"CV7BSTUW","created_at":"2026-07-05T11:38:58.674445+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/CV7BSTUWKFUGPCM2N6Y7MVGMYW","json":"https://pith.science/pith/CV7BSTUWKFUGPCM2N6Y7MVGMYW.json","graph_json":"https://pith.science/api/pith-number/CV7BSTUWKFUGPCM2N6Y7MVGMYW/graph.json","events_json":"https://pith.science/api/pith-number/CV7BSTUWKFUGPCM2N6Y7MVGMYW/events.json","paper":"https://pith.science/paper/CV7BSTUW"},"agent_actions":{"view_html":"https://pith.science/pith/CV7BSTUWKFUGPCM2N6Y7MVGMYW","download_json":"https://pith.science/pith/CV7BSTUWKFUGPCM2N6Y7MVGMYW.json","view_paper":"https://pith.science/paper/CV7BSTUW","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2507.13298&json=true","fetch_graph":"https://pith.science/api/pith-number/CV7BSTUWKFUGPCM2N6Y7MVGMYW/graph.json","fetch_events":"https://pith.science/api/pith-number/CV7BSTUWKFUGPCM2N6Y7MVGMYW/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/CV7BSTUWKFUGPCM2N6Y7MVGMYW/action/timestamp_anchor","attest_storage":"https://pith.science/pith/CV7BSTUWKFUGPCM2N6Y7MVGMYW/action/storage_attestation","attest_author":"https://pith.science/pith/CV7BSTUWKFUGPCM2N6Y7MVGMYW/action/author_attestation","sign_citation":"https://pith.science/pith/CV7BSTUWKFUGPCM2N6Y7MVGMYW/action/citation_signature","submit_replication":"https://pith.science/pith/CV7BSTUWKFUGPCM2N6Y7MVGMYW/action/replication_record"}},"created_at":"2026-07-05T11:38:58.674445+00:00","updated_at":"2026-07-05T11:38:58.674445+00:00"}