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Using a chain partition of an adequate finite poset we show that the Hausdorff dimension of an $n$-cube antichain is at most $n-1$.We conjecture that the $(n-1)$-dimensional Hausdorff measure of an $n$-cube antichain is at most $n$ times the Hausdorff measure of a facet of the unit $n$-cube and we verify this conjecture for $n=2$ as well as under the assumptio"},"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":"1707.04856","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-07-16T10:47:14Z","cross_cats_sorted":[],"title_canon_sha256":"a8862767447ce6ae869252a0e4bd133f549346150a69eeebc03503038fcd8c1d","abstract_canon_sha256":"63c228b7e7d579a74b0e6b9fd48205176c75406460c7a1c44288fdf840f1cd55"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:40:12.009026Z","signature_b64":"FFPk799WKQi61WRB19hRF4jlnNk3wd3pww1MqmDa0iAvWglC79vmGskiPHOKWXjQJr2bK46HcdI2yGdPX7BUCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4d86ae87dca0048ce4c5d7d3b9c0dcc95d0c9fc123bad331e67a4ea58a874d27","last_reissued_at":"2026-05-18T00:40:12.008300Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:40:12.008300Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A fractal perspective on optimal antichains and intersecting subsets of the unit $n$-cube","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Christos Pelekis, Konrad Engel, Themis Mitsis","submitted_at":"2017-07-16T10:47:14Z","abstract_excerpt":"An \\emph{$n$-cube antichain} is a subset of the unit $n$-cube $[0,1]^n$ that does not contain two elements $\\mathbf{x}=(x_1, x_2,\\ldots, x_n)$ and $\\mathbf{y}=(y_1, y_2,\\ldots, y_n)$ satisfying $x_i\\le y_i$ for all $i\\in \\{1,\\ldots,n\\}$. 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