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In this paper we show that, for $n\\geq k+l,l\\geq r\\geq 1,c>0$ and $\\mathscr{A}\\subseteq \\binom{[n]}{k},\\mathscr{B}\\subseteq \\binom{[n]}{l}$, if $\\mathscr{A}$ and $\\mathscr{B}$ are cross-intersecting and $\\binom{n-r}{l-r}\\leq|\\mathscr{B}|\\leq \\binom{n-1}{l-1}$, then $$|\\mathscr{A}|+c|\\mathscr{B}|\\leq \\max\\left\\{\\binom{n}{k}-\\bi"},"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":"2009.09396","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2020-09-20T09:33:11Z","cross_cats_sorted":[],"title_canon_sha256":"d5c000688f157047aa8bcf1392bc6edf1bc5f036e3390cbd89c7ab9cf8267772","abstract_canon_sha256":"708aca92333de34abda79eb3a325c3c22d22e83426573e0352cb9bf3ed63caae"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T01:41:09.631880Z","signature_b64":"fHnEbX5piFCT8FQaYpwG/r6EBQr1zWFoWobmCmvL/pJs4fAj4+Bntq/xhF+/eK1pharE9YobL2ib6yYcTdvNAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1b932342be7b7f1c3c5683964d48cfcd5709df47ea9e10908c97883025a6cb10","last_reissued_at":"2026-07-05T01:41:09.631478Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T01:41:09.631478Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On non-empty cross-intersecting families","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Chao Shi, Jianguo Qian, Peter Frankl","submitted_at":"2020-09-20T09:33:11Z","abstract_excerpt":"Let $2^{[n]}$ and $\\binom{[n]}{i}$ be the power set and the class of all $i$-subsets of $\\{1,2,\\cdots,n\\}$, respectively. 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