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Balogh, Liu and Sharifzadeh made significant progress on this question showing that it is $2^{O(r_k(n))}$ for an infinite sequence of $n$. We improve their result in two ways. On the one hand, we prove that, for $k\\geq 5$, the number of $k$-AP-free sets in $[n]$ is $2^{r_k(n)(1+o(1))}$ for an infinite sequence of $n$, solving the question of Cameron and Erd\\H{o}s for infinitely many value"},"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":"2607.17746","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.CO","submitted_at":"2026-07-20T09:40:09Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"99ae74538760458553171d5bf1b2e29da3a96fd3bfb0b70c670b6c9be524a1fc","abstract_canon_sha256":"c6f90c521837ccb07e7671d880fdf94cd24e98202b3ceb79ba41468f5058f30f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-21T02:21:57.805180Z","signature_b64":"ug1aeZG5866afBDjmS4Z/p8yWTCX9u+Dy803e2iUfKzEnQp7jf4z0LFpbylKnooDI6oOXN3i9I0ZklULPfYhBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e30a1a66598d32a1404ce2fd6949369c36a16bb7818d228cc8d1cb52521061f7","last_reissued_at":"2026-07-21T02:21:57.804314Z","signature_status":"signed_v1","first_computed_at":"2026-07-21T02:21:57.804314Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Counting subsets of integers free of arithmetic configurations","license":"http://creativecommons.org/licenses/by-sa/4.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Juanjo Ru\\'e, Miquel Ortega, Patrick Morris","submitted_at":"2026-07-20T09:40:09Z","abstract_excerpt":"Cameron and Erd\\H{o}s asked if the number of sets free of arithmetic progressions of length $k$ is $2^{r_k(n)(1+o(1))}$, where $r_k(n)$ is the maximum cardinality of a $k$-AP-free subset of $\\{1, \\dots, n\\}$. Balogh, Liu and Sharifzadeh made significant progress on this question showing that it is $2^{O(r_k(n))}$ for an infinite sequence of $n$. We improve their result in two ways. 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