{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:BIIXFIM2OXVJL7OGACJ5TP2SK3","short_pith_number":"pith:BIIXFIM2","canonical_record":{"source":{"id":"1711.07456","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-11-20T18:37:50Z","cross_cats_sorted":["math.AT","math.RT"],"title_canon_sha256":"3ddfd256f8c3998899111d0fefefb2befa24e5b4bdc762d86f6e422e1724545c","abstract_canon_sha256":"95c31184295e4e22072879e685d2082b92748427fd63858d7de46d20e682c466"},"schema_version":"1.0"},"canonical_sha256":"0a1172a19a75ea95fdc60093d9bf5256da272ce755d8e841e562174767c50129","source":{"kind":"arxiv","id":"1711.07456","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1711.07456","created_at":"2026-05-18T00:29:45Z"},{"alias_kind":"arxiv_version","alias_value":"1711.07456v2","created_at":"2026-05-18T00:29:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1711.07456","created_at":"2026-05-18T00:29:45Z"},{"alias_kind":"pith_short_12","alias_value":"BIIXFIM2OXVJ","created_at":"2026-05-18T12:31:08Z"},{"alias_kind":"pith_short_16","alias_value":"BIIXFIM2OXVJL7OG","created_at":"2026-05-18T12:31:08Z"},{"alias_kind":"pith_short_8","alias_value":"BIIXFIM2","created_at":"2026-05-18T12:31:08Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:BIIXFIM2OXVJL7OGACJ5TP2SK3","target":"record","payload":{"canonical_record":{"source":{"id":"1711.07456","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-11-20T18:37:50Z","cross_cats_sorted":["math.AT","math.RT"],"title_canon_sha256":"3ddfd256f8c3998899111d0fefefb2befa24e5b4bdc762d86f6e422e1724545c","abstract_canon_sha256":"95c31184295e4e22072879e685d2082b92748427fd63858d7de46d20e682c466"},"schema_version":"1.0"},"canonical_sha256":"0a1172a19a75ea95fdc60093d9bf5256da272ce755d8e841e562174767c50129","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:29:45.100679Z","signature_b64":"CKRsgIALUKe27AcReGiCINZW3Na3XX8CKmzdfobcPOONvPXasYlaqbmSaEWw8OSwg6ek1ugHprw2qYmC0uEfCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0a1172a19a75ea95fdc60093d9bf5256da272ce755d8e841e562174767c50129","last_reissued_at":"2026-05-18T00:29:45.100087Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:29:45.100087Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1711.07456","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:29:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"EnViXl+9jGArig24ESXLW7+W2aUELbeZ/EltxKjgFNCMgWfPD/adSbGdMPt9FVOpyuiYGVBUinwFbI9d+uLZCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T12:32:25.841071Z"},"content_sha256":"20bec5637ca8fd1e9b46323eda40ff50bbe60a231b31712f0389b110067119c6","schema_version":"1.0","event_id":"sha256:20bec5637ca8fd1e9b46323eda40ff50bbe60a231b31712f0389b110067119c6"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:BIIXFIM2OXVJL7OGACJ5TP2SK3","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Families of nested graphs with compatible symmetric-group actions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AT","math.RT"],"primary_cat":"math.CO","authors_text":"Eric Ramos, Graham White","submitted_at":"2017-11-20T18:37:50Z","abstract_excerpt":"For fixed positive integers $n$ and $k$, the Kneser graph $KG_{n,k}$ has vertices labeled by $k$-element subsets of $\\{1,2,\\dots,n\\}$ and edges between disjoint sets. Keeping $k$ fixed and allowing $n$ to grow, one obtains a family of nested graphs, each of which is acted on by a symmetric group in a way which is compatible with all of the other actions. In this paper, we provide a framework for studying families of this kind using the FI-module theory of Church, Ellenberg, and Farb, and show that this theory has a variety of asymptotic consequences for such families of graphs. These consequen"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.07456","kind":"arxiv","version":2},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:29:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FOKAciyZAyaypTQ+35GFH1S3JX3hDdnzH+swBsgc52Ci5Iqv/9DEhi7x9Ch/AK00FnTBKeTGnfPq0EUPT+EiBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T12:32:25.841761Z"},"content_sha256":"0352e641acfe4ee9afb19c6b46b04757301029be5aec92034a5ab255dfd7c136","schema_version":"1.0","event_id":"sha256:0352e641acfe4ee9afb19c6b46b04757301029be5aec92034a5ab255dfd7c136"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/BIIXFIM2OXVJL7OGACJ5TP2SK3/bundle.json","state_url":"https://pith.science/pith/BIIXFIM2OXVJL7OGACJ5TP2SK3/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/BIIXFIM2OXVJL7OGACJ5TP2SK3/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-05-26T12:32:25Z","links":{"resolver":"https://pith.science/pith/BIIXFIM2OXVJL7OGACJ5TP2SK3","bundle":"https://pith.science/pith/BIIXFIM2OXVJL7OGACJ5TP2SK3/bundle.json","state":"https://pith.science/pith/BIIXFIM2OXVJL7OGACJ5TP2SK3/state.json","well_known_bundle":"https://pith.science/.well-known/pith/BIIXFIM2OXVJL7OGACJ5TP2SK3/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:BIIXFIM2OXVJL7OGACJ5TP2SK3","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":"95c31184295e4e22072879e685d2082b92748427fd63858d7de46d20e682c466","cross_cats_sorted":["math.AT","math.RT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-11-20T18:37:50Z","title_canon_sha256":"3ddfd256f8c3998899111d0fefefb2befa24e5b4bdc762d86f6e422e1724545c"},"schema_version":"1.0","source":{"id":"1711.07456","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1711.07456","created_at":"2026-05-18T00:29:45Z"},{"alias_kind":"arxiv_version","alias_value":"1711.07456v2","created_at":"2026-05-18T00:29:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1711.07456","created_at":"2026-05-18T00:29:45Z"},{"alias_kind":"pith_short_12","alias_value":"BIIXFIM2OXVJ","created_at":"2026-05-18T12:31:08Z"},{"alias_kind":"pith_short_16","alias_value":"BIIXFIM2OXVJL7OG","created_at":"2026-05-18T12:31:08Z"},{"alias_kind":"pith_short_8","alias_value":"BIIXFIM2","created_at":"2026-05-18T12:31:08Z"}],"graph_snapshots":[{"event_id":"sha256:0352e641acfe4ee9afb19c6b46b04757301029be5aec92034a5ab255dfd7c136","target":"graph","created_at":"2026-05-18T00:29:45Z","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":"For fixed positive integers $n$ and $k$, the Kneser graph $KG_{n,k}$ has vertices labeled by $k$-element subsets of $\\{1,2,\\dots,n\\}$ and edges between disjoint sets. Keeping $k$ fixed and allowing $n$ to grow, one obtains a family of nested graphs, each of which is acted on by a symmetric group in a way which is compatible with all of the other actions. In this paper, we provide a framework for studying families of this kind using the FI-module theory of Church, Ellenberg, and Farb, and show that this theory has a variety of asymptotic consequences for such families of graphs. These consequen","authors_text":"Eric Ramos, Graham White","cross_cats":["math.AT","math.RT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-11-20T18:37:50Z","title":"Families of nested graphs with compatible symmetric-group actions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.07456","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":"sha256:20bec5637ca8fd1e9b46323eda40ff50bbe60a231b31712f0389b110067119c6","target":"record","created_at":"2026-05-18T00:29:45Z","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":"95c31184295e4e22072879e685d2082b92748427fd63858d7de46d20e682c466","cross_cats_sorted":["math.AT","math.RT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-11-20T18:37:50Z","title_canon_sha256":"3ddfd256f8c3998899111d0fefefb2befa24e5b4bdc762d86f6e422e1724545c"},"schema_version":"1.0","source":{"id":"1711.07456","kind":"arxiv","version":2}},"canonical_sha256":"0a1172a19a75ea95fdc60093d9bf5256da272ce755d8e841e562174767c50129","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0a1172a19a75ea95fdc60093d9bf5256da272ce755d8e841e562174767c50129","first_computed_at":"2026-05-18T00:29:45.100087Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:29:45.100087Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"CKRsgIALUKe27AcReGiCINZW3Na3XX8CKmzdfobcPOONvPXasYlaqbmSaEWw8OSwg6ek1ugHprw2qYmC0uEfCg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:29:45.100679Z","signed_message":"canonical_sha256_bytes"},"source_id":"1711.07456","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:20bec5637ca8fd1e9b46323eda40ff50bbe60a231b31712f0389b110067119c6","sha256:0352e641acfe4ee9afb19c6b46b04757301029be5aec92034a5ab255dfd7c136"],"state_sha256":"0d5d5b5e683610e462b2d5f8e474764f1c6907fed616b11c486abc0608b485ba"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ZWNXXT5LgkefkuvkD04qetKBlmca5lnzmBvU7P3Tc8KYwYqFaiMlnAroOQkXnXj7aDcuZMglYzZZjXsgEaXmBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-26T12:32:25.845243Z","bundle_sha256":"c661df1a7391bd43f84471ae6bd5d01ee299dfacdd2e9e507c711f98c400c380"}}