A Fano-plane counterexample to a conjecture on intersecting multipartite hypergraphs

Abstract

We construct a seven-partite seven-uniform hypergraph with 28 vertices and 22 edges in which every two edges meet in at least two vertices and the ordinary vertex cover number is four. The construction is based on the seven lines of the Fano plane and has a cyclic description by three words of length seven. An elementary proof excludes every three-vertex cover. This disproves Conjecture 3.1 of Bishnoi, Das, Morris and Szabó at (r,t)=(7,2). Together with their reported upper bound it yields Ryser(7,2)=4. The construction is edge-critical and uses the minimum possible number of vertices. The record includes the manuscript, LaTeX source, explicit edge and avoidance certificates, and independent exhaustive verification programs. OpenAI Codex assisted with mathematical exploration, computational verification and manuscript preparation. The author is responsible for the final manuscript and accompanying materials. License scope: manuscript, documentation and original mathematical certificate data are CC BY 4.0; original code is Apache-2.0. Third-party materials retain their own attribution and licenses. Version 1.0.2 implements a new independent AI review-revision-cross-review cycle, with two reviews per manuscript. All adopted comments were resolved in the second review. See REVISION_NOTES.txt and review_history/ in the complete source archive. The main result and mathematical certificate data are unchanged. This is independent AI review, not external human journal peer review. Published version 1.0.1 remains available separately.