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We greatly extend this by showing that if $G$ has average degree $d$, then $\\mbox{disc}^{+}(G)=\\Omega(d^{\\frac{1}{2}}n)$ if $d\\in [0,n^{\\frac{2}{3}}]$, $\\Omega(n^2/d)$ if $d\\in [n^{\\frac{2}{3}},n^{\\frac{4}{5}}]$, and $\\Omega(d^{\\frac{1}{4}}n/\\log "},"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":"2311.02070","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2023-11-03T17:54:27Z","cross_cats_sorted":[],"title_canon_sha256":"7bbe605577d837b75e542f3bbfec0a282b046d50cb9d9dec82ce8f1c7752dab2","abstract_canon_sha256":"9ace5bf9b60f2aa91cb0162909d13f644ce450bde60066cfa02de0ddcf1b2b6b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T07:14:20.046360Z","signature_b64":"woFtv/2iHQMKFaTmlWJLOhhJTWi+JYlVkp4LQJxnqzGt3dtatqNKYBL9Wo1yvR1YRbxJwNFEv9dlXeShSy6DBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"abbe671b6dabbaab8be1cad555da2f6d411075384067a4f09d4e4ce45b9259af","last_reissued_at":"2026-07-05T07:14:20.045912Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T07:14:20.045912Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Positive discrepancy, MaxCut, and eigenvalues of graphs","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Benny Sudakov, Eero R\\\"aty, Istv\\'an Tomon","submitted_at":"2023-11-03T17:54:27Z","abstract_excerpt":"The positive discrepancy of a graph $G$ of edge density $p=e(G)/\\binom{v(G)}{2}$ is defined as\n  $$\\mbox{disc}^{+}(G)=\\max_{U\\subset V(G)}e(G[U])-p\\binom{|U|}{2}.$$\n  In 1993, Alon proved (using the equivalent terminology of minimum bisections) that if $G$ is $d$-regular on $n$ vertices, and $d=O(n^{1/9})$, then $\\mbox{disc}^{+}(G)=\\Omega(d^{1/2}n)$. 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