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In this paper we consider the following three general questions:\n  (i) What is the size of the largest $\\mathcal{L}$-free subset of $[n]$?\n  (ii) How many $\\mathcal{L}$-free subsets of $[n]$ are there?\n  (iii) How many maximal $\\mathcal{L}$-free subsets of $[n]$ are there?\n  We completely resolve (i) in the case when $\\mathcal{L}$ is the equation $px+qy=z$ for fixed $p,q\\in \\mathbb N$ where $p\\geq 2$. Further, up to a multiplicative constant"},"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":"1607.08399","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-07-28T11:07:18Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"b6d5397e138912ce951fdb74e938397c959bc93fea0fc46dcf0453d9d4497dd2","abstract_canon_sha256":"1c798efd581278037c8d08ce6f7cda3e1481ba950fd9cd3c4a5cce205e2e09b5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:01:51.579229Z","signature_b64":"Amatn5E3Hcf5MWyu289M2mueR/ANO4vMT6MPHugCpANGjBfv7KCpPANurlrO9FZNzm/Xn0fc1j88OuNIVrKjAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"10bf0a1d2c6b1838a6af41f9c2a129ca37c67dafdcfa76ad8b2d9dafec5598d1","last_reissued_at":"2026-05-18T01:01:51.578573Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:01:51.578573Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On solution-free sets of integers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Andrew Treglown, Robert Hancock","submitted_at":"2016-07-28T11:07:18Z","abstract_excerpt":"Given a linear equation $\\mathcal{L}$, a set $A \\subseteq [n]$ is $\\mathcal{L}$-free if $A$ does not contain any `non-trivial' solutions to $\\mathcal{L}$. 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