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For similar conditions and sufficiently large $n$, (inspired by the work of P. D. T. A. Elliott in \\cite{Ell67}) we also show that the number of spheres determined by $n$ points is at least $1+\\binom{n-1}{3}-t_3^{orchard}(n-1)$, and this bound is best possible under its hypothesis. (By $t_3^{orchard}(n)$, we are denoting the maximum number of t"},"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":"0907.0724","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2009-07-03T22:11:16Z","cross_cats_sorted":[],"title_canon_sha256":"88380ca7ec8b28858adb506b73326b81b4cfea8ffa0ae147a29b27aaf5f4ce8a","abstract_canon_sha256":"1fdf04212992a601c5a0d06017327bce1f5daea86355ab2986f391b97c271473"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:39:39.168014Z","signature_b64":"hu6TzKnTrwU6vrpwyUX64+t25QXT9A/x2rWKQ0Ev5PXi4WufrfThbaBQ/Nz1hVZn+weQOEglYosRQQRGWGcRAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"de5a3938f10a23c7cc4c9ad96aa0cf30cd32afb1e3c84ced93f42fceddd88765","last_reissued_at":"2026-05-18T04:39:39.167533Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:39:39.167533Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Lines, Circles, Planes and Spheres","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"George B. Purdy, Justin W. Smith","submitted_at":"2009-07-03T22:11:16Z","abstract_excerpt":"Let $S$ be a set of $n$ points in $\\mathbb{R}^3$, no three collinear and not all coplanar. If at most $n-k$ are coplanar and $n$ is sufficiently large, the total number of planes determined is at least $1 + k \\binom{n-k}{2}-\\binom{k}{2}(\\frac{n-k}{2})$. For similar conditions and sufficiently large $n$, (inspired by the work of P. D. T. A. Elliott in \\cite{Ell67}) we also show that the number of spheres determined by $n$ points is at least $1+\\binom{n-1}{3}-t_3^{orchard}(n-1)$, and this bound is best possible under its hypothesis. 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