{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:I7ELAA7S7ACG2QZTJX6PBBTL5S","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"bd079a2240682305a508d5b23f34b68e430abf00b7b6b04f0e089f9dad338623","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-04-28T01:15:18Z","title_canon_sha256":"6f801d8e27032e34f464826744bf842ed38287b9938483520326f3c9ea3f13cc"},"schema_version":"1.0","source":{"id":"1804.10717","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1804.10717","created_at":"2026-05-18T00:05:04Z"},{"alias_kind":"arxiv_version","alias_value":"1804.10717v2","created_at":"2026-05-18T00:05:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1804.10717","created_at":"2026-05-18T00:05:04Z"},{"alias_kind":"pith_short_12","alias_value":"I7ELAA7S7ACG","created_at":"2026-05-18T12:32:28Z"},{"alias_kind":"pith_short_16","alias_value":"I7ELAA7S7ACG2QZT","created_at":"2026-05-18T12:32:28Z"},{"alias_kind":"pith_short_8","alias_value":"I7ELAA7S","created_at":"2026-05-18T12:32:28Z"}],"graph_snapshots":[{"event_id":"sha256:7d684b2eb044b7108f66fdf99d08b522b65d16a2b5ef3250474ee9700d32c4ab","target":"graph","created_at":"2026-05-18T00:05:04Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"Let $\\text{Tr}(n,m,k)$ denote the largest number of distinct projections onto $k$ coordinates guaranteed in any family of $m$ binary vectors of length $n$. The classical Sauer-Perles-Shelah Lemma implies that $\\text{Tr}(n, n^r, k) = 2^k$ for $k \\le r$. While determining $\\text{Tr}(n,n^r,k)$ precisely for general $k$ seems hopeless even for constant $r$, estimating it, and more generally estimating the function $\\text{Tr}(n,m,k)$ for all range of the parameters, remains a widely open problem with connections to important questions in computer science and combinatorics. Here we essentially resol","authors_text":"Guy Moshkovitz, Noam Solomon, Noga Alon","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-04-28T01:15:18Z","title":"Traces of Hypergraphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.10717","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:ecf141825e76f9c72e7e1c61aba7ac2e20a48c153272c5faa15f6dc19d6c9a26","target":"record","created_at":"2026-05-18T00:05:04Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"bd079a2240682305a508d5b23f34b68e430abf00b7b6b04f0e089f9dad338623","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-04-28T01:15:18Z","title_canon_sha256":"6f801d8e27032e34f464826744bf842ed38287b9938483520326f3c9ea3f13cc"},"schema_version":"1.0","source":{"id":"1804.10717","kind":"arxiv","version":2}},"canonical_sha256":"47c8b003f2f8046d43334dfcf0866beca1bb86dece8d62d80b028f52248b3ee4","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"47c8b003f2f8046d43334dfcf0866beca1bb86dece8d62d80b028f52248b3ee4","first_computed_at":"2026-05-18T00:05:04.998588Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:05:04.998588Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"QbXX88xMJC95dGXnjwS0awM6/Iyeh+dtSx1FJgYke7pWfLa5lCr64csID49tKv//WXaIJMqgLsyDrrg4OEFnAw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:05:04.999271Z","signed_message":"canonical_sha256_bytes"},"source_id":"1804.10717","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ecf141825e76f9c72e7e1c61aba7ac2e20a48c153272c5faa15f6dc19d6c9a26","sha256:7d684b2eb044b7108f66fdf99d08b522b65d16a2b5ef3250474ee9700d32c4ab"],"state_sha256":"a38cf825d4484009dd8db8bc96e474079f712deed40d98e337f6fc3d51be0ad2"}