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We refer to such a curve $A$ as an $n$-\\emph{de~Bruijn-Erd\\H{o}s-set}. Under the additional assumption that all functions $f_i,i=1,\\ldots,n-2,$ are piecewise monotone, we show that the Hausdorff dimension of $A$ is at most $1$ as wel"},"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":"1805.10980","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2018-05-28T15:39:32Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"46ef82d1dd846ae76a3272341e1f12007573389bfbd7b2dfea9336026337b704","abstract_canon_sha256":"0f8d4b0e5900f21f52bb613b680279dfd559b294b946edb0b525a1340c78fd56"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:14:48.642301Z","signature_b64":"kjlskTWPaZebpCMw0eZCbRjKvsMtbAEonto82iZTeYGj7F8dODrp6v/xq+LaMZwWr3RfYbIAtZZSh3elAcSRCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"aaa3074203415a8f8ce6235315c67fa7a3622d1ee7928b9d69a3934035d39adc","last_reissued_at":"2026-05-18T00:14:48.641603Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:14:48.641603Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The de Bruijn-Erd\\H{o}s theorem from a Hausdorff measure point of view","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.CA","authors_text":"Christos Pelekis, Martin Dole\\v{z}al, Themis Mitsis","submitted_at":"2018-05-28T15:39:32Z","abstract_excerpt":"Motivated by a well-known result in extremal set theory, due to Nicolaas Govert de Bruijn and Paul Erd\\H{o}s, we consider curves in the unit $n$-cube $[0,1]^n$ of the form \\[ A=\\{(x,f_1(x),\\ldots,f_{n-2}(x),\\alpha): x\\in [0,1]\\}, \\] where $\\alpha$ is a fixed real number in $[0,1]$ and $f_1,\\ldots,f_{n-2}$ are injective measurable functions from $[0,1]$ to $[0,1]$. 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Under the additional assumption that all functions $f_i,i=1,\\ldots,n-2,$ are piecewise monotone, we show that the Hausdorff dimension of $A$ is at most $1$ as wel"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.10980","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":"1805.10980","created_at":"2026-05-18T00:14:48.641709+00:00"},{"alias_kind":"arxiv_version","alias_value":"1805.10980v1","created_at":"2026-05-18T00:14:48.641709+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1805.10980","created_at":"2026-05-18T00:14:48.641709+00:00"},{"alias_kind":"pith_short_12","alias_value":"VKRQOQQDIFNI","created_at":"2026-05-18T12:32:59.047623+00:00"},{"alias_kind":"pith_short_16","alias_value":"VKRQOQQDIFNI7DHG","created_at":"2026-05-18T12:32:59.047623+00:00"},{"alias_kind":"pith_short_8","alias_value":"VKRQOQQD","created_at":"2026-05-18T12:32:59.047623+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/VKRQOQQDIFNI7DHGENJRLRT7U6","json":"https://pith.science/pith/VKRQOQQDIFNI7DHGENJRLRT7U6.json","graph_json":"https://pith.science/api/pith-number/VKRQOQQDIFNI7DHGENJRLRT7U6/graph.json","events_json":"https://pith.science/api/pith-number/VKRQOQQDIFNI7DHGENJRLRT7U6/events.json","paper":"https://pith.science/paper/VKRQOQQD"},"agent_actions":{"view_html":"https://pith.science/pith/VKRQOQQDIFNI7DHGENJRLRT7U6","download_json":"https://pith.science/pith/VKRQOQQDIFNI7DHGENJRLRT7U6.json","view_paper":"https://pith.science/paper/VKRQOQQD","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1805.10980&json=true","fetch_graph":"https://pith.science/api/pith-number/VKRQOQQDIFNI7DHGENJRLRT7U6/graph.json","fetch_events":"https://pith.science/api/pith-number/VKRQOQQDIFNI7DHGENJRLRT7U6/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/VKRQOQQDIFNI7DHGENJRLRT7U6/action/timestamp_anchor","attest_storage":"https://pith.science/pith/VKRQOQQDIFNI7DHGENJRLRT7U6/action/storage_attestation","attest_author":"https://pith.science/pith/VKRQOQQDIFNI7DHGENJRLRT7U6/action/author_attestation","sign_citation":"https://pith.science/pith/VKRQOQQDIFNI7DHGENJRLRT7U6/action/citation_signature","submit_replication":"https://pith.science/pith/VKRQOQQDIFNI7DHGENJRLRT7U6/action/replication_record"}},"created_at":"2026-05-18T00:14:48.641709+00:00","updated_at":"2026-05-18T00:14:48.641709+00:00"}