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While the conjecture was verified for $k \\leq 9$ by Gao et al., it was disproved by Huang and Sudakov, and further Balodis et al. proved that $f(k) \\geq 2^{\\widetilde{\\Omega}((\\log k)^2)}$.\n  In this note, we give a simple proof of the recursive upper bound $f(k+1) \\leq f(k)+f(\\lfloor k/4 \\rfloor)$. Consequently, $f(k) \\leq 2^{(\\log_2 (4k))^2/4}$ for $k \\geq 1$. This improves the previous be"},"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":"2605.28915","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-05-27T17:50:03Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"92dad904e5eb92a72cb9d719738189551e10e6421077e75cf32ceec46a75b396","abstract_canon_sha256":"85d33a75c881b4d64a551536afd8df71d1e69af61d8b682da259532c27c8db3c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-29T00:04:16.435633Z","signature_b64":"+eojH/t9T6Sx03IizC0fbbeb/P55apC4BaBOSV0PCvCM0IFxd/5hh0mji4fn5dJPSTOY0wIT5AV4XPozyCWSBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"35212a5ea36d5bf348cdb03de981634753aeb46f159a11ddbd9ff6b0237a6961","last_reissued_at":"2026-05-29T00:04:16.435081Z","signature_status":"signed_v1","first_computed_at":"2026-05-29T00:04:16.435081Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A note on the Alon-Saks-Seymour problem","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Jacob Fox","submitted_at":"2026-05-27T17:50:03Z","abstract_excerpt":"Let $f(k)$ be the maximum possible chromatic number of a graph whose edge set can be partitioned into at most $k$ complete bipartite graphs. Alon, Saks, and Seymour conjectured that $f(k)=k+1$ for all $k$. While the conjecture was verified for $k \\leq 9$ by Gao et al., it was disproved by Huang and Sudakov, and further Balodis et al. proved that $f(k) \\geq 2^{\\widetilde{\\Omega}((\\log k)^2)}$.\n  In this note, we give a simple proof of the recursive upper bound $f(k+1) \\leq f(k)+f(\\lfloor k/4 \\rfloor)$. Consequently, $f(k) \\leq 2^{(\\log_2 (4k))^2/4}$ for $k \\geq 1$. 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