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For every $i\\le n$, the $n$-avoidance basis contains $C_i$. Clark showed that the avoidability index of every circular formula and of every formula in the $3$-avoidance basis (and thus of every avoidable formula containing at most 3 variables) is at most 4. We determine exactly the avoidability index of these formulas."},"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":"1610.04439","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2016-10-14T12:53:38Z","cross_cats_sorted":[],"title_canon_sha256":"1ff2bed3e6182030430a3e012e2ecae39203159c2abc0cd9b40bf766ec365fc1","abstract_canon_sha256":"f3a0f648fd6649805129e34e13fe700110559548666c80903e69015386a887a1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:02:15.197853Z","signature_b64":"IhdjZdWh1DQJtPNV0ChHU8vFgmhhb2xlL0zAPKnV4NT/sV9o6LifPjNWjWWXvD/46PH43qo2GRHGY8e/7wAKDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8209dcb34096b4f80306af66c8844f76551f0a6af96a0aa0dd698e36b3ca40e4","last_reissued_at":"2026-05-18T01:02:15.197132Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:02:15.197132Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Avoidability of circular formulas","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.DM","authors_text":"Guilhem Gamard, Gwena\\\"el Richomme, Pascal Ochem, Patrice S\\'e\\'ebold","submitted_at":"2016-10-14T12:53:38Z","abstract_excerpt":"Clark has defined the notion of $n$-avoidance basis which contains the avoidable formulas with at most $n$ variables that are closest to be unavoidable in some sense. The family $C_i$ of circular formulas is such that $C_1=AA$, $C_2=ABA.BAB$, $C_3=ABCA.BCAB.CABC$ and so on. For every $i\\le n$, the $n$-avoidance basis contains $C_i$. Clark showed that the avoidability index of every circular formula and of every formula in the $3$-avoidance basis (and thus of every avoidable formula containing at most 3 variables) is at most 4. 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