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We denote by $\\phi(x,y)$ the maximum $z$ such that, in all such graphs $G$, there is a vertex $v \\in C$ that is joined to at least $z|A|$ vertices in $A$ by two-edge paths. If in addition we require that every vertex in $B$ has at least $x|A|$ neighbors in $A$, and every vertex in $C$ has at least $y|B|$ neighbors in $C$, we d"},"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":"1908.07453","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-08-19T01:06:42Z","cross_cats_sorted":[],"title_canon_sha256":"19f7cf90597e317e1d69e7f58c91ce4d8d8a581488f73123f76ce344cb7133d0","abstract_canon_sha256":"9a51a6a7af9f45bb82b0670a2e0f46c8109c1592b0fd1c5aeb28bc2db792c928"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T04:34:25.023723Z","signature_b64":"tqOPJ6HGYNg2luuT8/dVBOM2ekNqDomBoxYAytkRAboOYP2KtjgtNE2aFk8VLwEKVm9B2SmykcDIibK7A6kOBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5884f6d1b86b118aa57cc2221d4345b7fec27800daa6e2e9b10130eb7d442e59","last_reissued_at":"2026-07-05T04:34:25.023351Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T04:34:25.023351Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Some results on concatenating bipartite graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Patrick Hompe","submitted_at":"2019-08-19T01:06:42Z","abstract_excerpt":"We consider two functions $\\phi$ and $\\psi$, defined as follows. Let $x,y \\in (0,1]$ and let $A,B,C$ be disjoint nonempty subsets of a graph $G$, where every vertex in $A$ has at least $x|B|$ neighbors in $B$, and every vertex in $B$ has at least $y|C|$ neighbors in $C$. We denote by $\\phi(x,y)$ the maximum $z$ such that, in all such graphs $G$, there is a vertex $v \\in C$ that is joined to at least $z|A|$ vertices in $A$ by two-edge paths. If in addition we require that every vertex in $B$ has at least $x|A|$ neighbors in $A$, and every vertex in $C$ has at least $y|B|$ neighbors in $C$, we d"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.07453","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1908.07453/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"1908.07453","created_at":"2026-07-05T04:34:25.023413+00:00"},{"alias_kind":"arxiv_version","alias_value":"1908.07453v3","created_at":"2026-07-05T04:34:25.023413+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.07453","created_at":"2026-07-05T04:34:25.023413+00:00"},{"alias_kind":"pith_short_12","alias_value":"LCCPNUNYNMIY","created_at":"2026-07-05T04:34:25.023413+00:00"},{"alias_kind":"pith_short_16","alias_value":"LCCPNUNYNMIYVJL4","created_at":"2026-07-05T04:34:25.023413+00:00"},{"alias_kind":"pith_short_8","alias_value":"LCCPNUNY","created_at":"2026-07-05T04:34:25.023413+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/LCCPNUNYNMIYVJL4YIRB2Q2FW7","json":"https://pith.science/pith/LCCPNUNYNMIYVJL4YIRB2Q2FW7.json","graph_json":"https://pith.science/api/pith-number/LCCPNUNYNMIYVJL4YIRB2Q2FW7/graph.json","events_json":"https://pith.science/api/pith-number/LCCPNUNYNMIYVJL4YIRB2Q2FW7/events.json","paper":"https://pith.science/paper/LCCPNUNY"},"agent_actions":{"view_html":"https://pith.science/pith/LCCPNUNYNMIYVJL4YIRB2Q2FW7","download_json":"https://pith.science/pith/LCCPNUNYNMIYVJL4YIRB2Q2FW7.json","view_paper":"https://pith.science/paper/LCCPNUNY","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1908.07453&json=true","fetch_graph":"https://pith.science/api/pith-number/LCCPNUNYNMIYVJL4YIRB2Q2FW7/graph.json","fetch_events":"https://pith.science/api/pith-number/LCCPNUNYNMIYVJL4YIRB2Q2FW7/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/LCCPNUNYNMIYVJL4YIRB2Q2FW7/action/timestamp_anchor","attest_storage":"https://pith.science/pith/LCCPNUNYNMIYVJL4YIRB2Q2FW7/action/storage_attestation","attest_author":"https://pith.science/pith/LCCPNUNYNMIYVJL4YIRB2Q2FW7/action/author_attestation","sign_citation":"https://pith.science/pith/LCCPNUNYNMIYVJL4YIRB2Q2FW7/action/citation_signature","submit_replication":"https://pith.science/pith/LCCPNUNYNMIYVJL4YIRB2Q2FW7/action/replication_record"}},"created_at":"2026-07-05T04:34:25.023413+00:00","updated_at":"2026-07-05T04:34:25.023413+00:00"}