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Paper 2. The Eleven-Card Lemma
Every eleven-card family has a four-cover, and fifteen cards then take two lines
Mohamed A. Osman · ORCID 0009-0004-5912-999X · independent researcher · 2026-10-19
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Abstract. Every pairwise intersecting 6-uniform family of eleven cards has a transversal of size at most four. The proof splits on the maximum degree: five or more closes by pairing, four by the K7 lemma (proved by hand in Paper 1), and three or less by a hand reduction to a short statement on eleven points, checked by a SAT solver in seconds, with the earlier search over 5,373 instances kept as a second check. The lemma also pins the maximum degree at exactly four in the sixteen-card stage. An explicit eight-card family shows the analogue fails at eight.
No priority is claimed for any result, and this is not presented as a settled answer. The lower bound is computer-assisted and has not been independently reviewed by a human or by a proof assistant. AI tools were used throughout the work and its review; every computational claim is regenerated by a script or certified by a proof file in code/ and results/ of the repository. Licence: CC BY 4.0 (text), MIT (code).