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Avis, Erd\\H{o}s and Pach (1988) introduced the extremal quantity $f_3(n)=\\max\\sum_{x\\in S}e_r(x,S)$, where the maximum is taken over all $n$-point subsets $S$ of 3-space and all assignments $r\\colon S\\to(0,\\infty)$ of distances. We show that if the pair $(S,r)$ maximises $f_3(n)$ and $n$ is sufficiently large, then, except for at most $2$ points, $S$ is contained in a circle $\\mathcal{C}$ and the axis of symmetry "},"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":"1907.08402","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-07-19T08:23:21Z","cross_cats_sorted":["cs.CG","math.MG"],"title_canon_sha256":"135b78e4cff1313dad8b9dbfe2e5323bcf64e8c467fa5a47888ef38196f0ca94","abstract_canon_sha256":"4853675e1ab2409a2dfb6d953bc13f77fd67c2afe68712c01fbc839ddea49325"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:40:09.964364Z","signature_b64":"KnyArp1dhn5EW0dIwG1+7rs810kaGXSnPcQpykEqdifjUYJKoyKxdjpkTS2EKKUoeg7690nSiTHLmyHHCPN1BQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c79af0b5e79ef7978aa15dcefb3eed398a02e7fbb715a4980408341bc416953a","last_reissued_at":"2026-05-17T23:40:09.963874Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:40:09.963874Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Favourite distances in 3-space","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CG","math.MG"],"primary_cat":"math.CO","authors_text":"Konrad J. Swanepoel","submitted_at":"2019-07-19T08:23:21Z","abstract_excerpt":"Let $S$ be a set of $n$ points in Euclidean $3$-space. Assign to each $x\\in S$ a distance $r(x)>0$, and let $e_r(x,S)$ denote the number of points in $S$ at distance $r(x)$ from $x$. Avis, Erd\\H{o}s and Pach (1988) introduced the extremal quantity $f_3(n)=\\max\\sum_{x\\in S}e_r(x,S)$, where the maximum is taken over all $n$-point subsets $S$ of 3-space and all assignments $r\\colon S\\to(0,\\infty)$ of distances. We show that if the pair $(S,r)$ maximises $f_3(n)$ and $n$ is sufficiently large, then, except for at most $2$ points, $S$ is contained in a circle $\\mathcal{C}$ and the axis of symmetry "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1907.08402","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1907.08402","created_at":"2026-05-17T23:40:09.963944+00:00"},{"alias_kind":"arxiv_version","alias_value":"1907.08402v1","created_at":"2026-05-17T23:40:09.963944+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1907.08402","created_at":"2026-05-17T23:40:09.963944+00:00"},{"alias_kind":"pith_short_12","alias_value":"Y6NPBNPHT33Z","created_at":"2026-05-18T12:33:33.725879+00:00"},{"alias_kind":"pith_short_16","alias_value":"Y6NPBNPHT33ZPCVB","created_at":"2026-05-18T12:33:33.725879+00:00"},{"alias_kind":"pith_short_8","alias_value":"Y6NPBNPH","created_at":"2026-05-18T12:33:33.725879+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/Y6NPBNPHT33ZPCVBLXHPWPXNHG","json":"https://pith.science/pith/Y6NPBNPHT33ZPCVBLXHPWPXNHG.json","graph_json":"https://pith.science/api/pith-number/Y6NPBNPHT33ZPCVBLXHPWPXNHG/graph.json","events_json":"https://pith.science/api/pith-number/Y6NPBNPHT33ZPCVBLXHPWPXNHG/events.json","paper":"https://pith.science/paper/Y6NPBNPH"},"agent_actions":{"view_html":"https://pith.science/pith/Y6NPBNPHT33ZPCVBLXHPWPXNHG","download_json":"https://pith.science/pith/Y6NPBNPHT33ZPCVBLXHPWPXNHG.json","view_paper":"https://pith.science/paper/Y6NPBNPH","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1907.08402&json=true","fetch_graph":"https://pith.science/api/pith-number/Y6NPBNPHT33ZPCVBLXHPWPXNHG/graph.json","fetch_events":"https://pith.science/api/pith-number/Y6NPBNPHT33ZPCVBLXHPWPXNHG/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/Y6NPBNPHT33ZPCVBLXHPWPXNHG/action/timestamp_anchor","attest_storage":"https://pith.science/pith/Y6NPBNPHT33ZPCVBLXHPWPXNHG/action/storage_attestation","attest_author":"https://pith.science/pith/Y6NPBNPHT33ZPCVBLXHPWPXNHG/action/author_attestation","sign_citation":"https://pith.science/pith/Y6NPBNPHT33ZPCVBLXHPWPXNHG/action/citation_signature","submit_replication":"https://pith.science/pith/Y6NPBNPHT33ZPCVBLXHPWPXNHG/action/replication_record"}},"created_at":"2026-05-17T23:40:09.963944+00:00","updated_at":"2026-05-17T23:40:09.963944+00:00"}