{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:PWZAMCCJJF36HNS4VP2AAC77AE","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":"a5a7d3502a322bc5933570d77d82f84a976b5b9d44812c5bb1603c8159cb0f25","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-08-06T16:46:05Z","title_canon_sha256":"0874872e1587814031fbb01b5d2ba1a33a3ad2aed9a1dfcbb2623574bfe0b14a"},"schema_version":"1.0","source":{"id":"1908.02253","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1908.02253","created_at":"2026-07-05T01:42:12Z"},{"alias_kind":"arxiv_version","alias_value":"1908.02253v2","created_at":"2026-07-05T01:42:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.02253","created_at":"2026-07-05T01:42:12Z"},{"alias_kind":"pith_short_12","alias_value":"PWZAMCCJJF36","created_at":"2026-07-05T01:42:12Z"},{"alias_kind":"pith_short_16","alias_value":"PWZAMCCJJF36HNS4","created_at":"2026-07-05T01:42:12Z"},{"alias_kind":"pith_short_8","alias_value":"PWZAMCCJ","created_at":"2026-07-05T01:42:12Z"}],"graph_snapshots":[{"event_id":"sha256:925f5158c516fc1d21b44b15910edc2f868282832e1791fafd810784392a8c4b","target":"graph","created_at":"2026-07-05T01:42:12Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/1908.02253/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"For a set $A\\subseteq\\left[k\\right]^{n}=\\left\\{ 0,\\dots,k-1\\right\\} ^{n}$, we define the $d$-shadow of $A$ to be the set of points obtained by flipping to zero one of the non-zero coordinates of some point in $A$. Let $\\left[k\\right]_{r}^{n}$ be the set of those points in $\\left[k\\right]^{n}$ with exactly $r$ non-zero coordinates. Given the size of $A$, how should we choose $A\\subseteq\\left[k\\right]_{r}^{n}$ so as to minimise the $d$-shadow? Note that the case $k=2$ is answered by the Kruskal-Katona theorem.\n  Our aim in this paper is to give an exact answer to this question. In particular, we","authors_text":"Eero Raty","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-08-06T16:46:05Z","title":"A grid generalisation of the Kruskal-Katona theorem"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.02253","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":"sha256:615302017c8927e622f4297131e842cf19a5309e9a218e2463793937682c04d6","target":"record","created_at":"2026-07-05T01:42:12Z","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":"a5a7d3502a322bc5933570d77d82f84a976b5b9d44812c5bb1603c8159cb0f25","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-08-06T16:46:05Z","title_canon_sha256":"0874872e1587814031fbb01b5d2ba1a33a3ad2aed9a1dfcbb2623574bfe0b14a"},"schema_version":"1.0","source":{"id":"1908.02253","kind":"arxiv","version":2}},"canonical_sha256":"7db20608494977e3b65cabf4000bff0130e79d128eadd16aafaa6d9e131112f6","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7db20608494977e3b65cabf4000bff0130e79d128eadd16aafaa6d9e131112f6","first_computed_at":"2026-07-05T01:42:12.527598Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T01:42:12.527598Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"T26M1iQcoLWvP34OUY+jkmaXjeSb8H0RJuOjX2gSR1Vz+/8XbLTeu4+JqG9YkYMv6JC5EsoCAq/51/bN7W5lBA==","signature_status":"signed_v1","signed_at":"2026-07-05T01:42:12.528021Z","signed_message":"canonical_sha256_bytes"},"source_id":"1908.02253","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:615302017c8927e622f4297131e842cf19a5309e9a218e2463793937682c04d6","sha256:925f5158c516fc1d21b44b15910edc2f868282832e1791fafd810784392a8c4b"],"state_sha256":"91a787b460a3f1229026d994a772c87a1b46db02d5521ef2208ab57819c1b8aa"}