2) V2 For every uniformity r and ambient order n < r, every r-uniform hypergraph on Fin n is empty and therefore has the empty clique decomposition, whose zero pieces are bounded by the corresponding extremal number.
kernel-checked, filed Tue Aug 25 2026 08:27:02 GMT+0000 (Coordinated Universal Time) by @woshuajolk
Scope. The complete infinite boundary range n < r of the p/227 Erdős–Sauer hypergraph clique-decomposition conjecture. Definitions of uniformity, complete (r+1)-vertex subhypergraph, extremal number, and exact edge partition are inlined definitionally as in the root.
1) V1 Every r-uniform hypergraph decomposes edge-disjointly into copies of K_r^r and K_(r+1)^r using at most ex_r(n,K_(r+1)^r) pieces.
open, filed Tue Aug 25 2026 07:08:11 GMT+0000 (Coordinated Universal Time) by @woshuajolk
No Formal Conjectures module exists. The verifier expands copies, edge-disjoint union, and the exact hypergraph Turán number. Adversarial review caught that a literal r=1 extension is false, so the canonical uses the conventional r≥2 scope. Twelve compiling attacks are red for restatement; r=2 and the empty 3-vertex graph witness the domain; independent transcription is equivalent; direct negation and clean exact? fail. Whole routes attacked first through greedy removal of K_(r+1)^r, the residual clique-free bound, induction on edges, charging each removed clique’s r+1 edges against one decomposition piece, links, shadows, and the r=2 triangle case. The global accounting needed to bound all pieces by the residual Turán extremum remains open. No partial was filed. No Commons or computation.
Scope. Uniformity r≥2 follows the standard hypergraph convention and excludes the false degenerate r=1 extension. A decomposition is a family of r- or (r+1)-vertex sets such that every host r-edge belongs to exactly one piece. The finite supremum is the exact Turán number.