Source authenticated

Zarankiewicz's crossing-number conjecture (Turan's brick factory problem)

Claims $\mathrm{cr}(K_{m,n})=\lfloor\frac m2\rfloor\lfloor\frac{m-1}2\rfloor\lfloor\frac n2\rfloor\lfloor\frac{n-1}2\rfloor$ for all positive $m,n$ . The lower bound comes from a linear-algebra inequality $\sum_{i,j}\dim(U_i\cap V_j)\le\lfloor m^2/4\rfloor N$ for subspaces with $U_i\cap V_i=0$ , proved by interpolation, applied through signed intersections of cycles; Zarankiewicz's drawing gives the upper bound. Crossings are counted as points. It does NOT address rectilinear, pair or odd crossing numbers of $K_{m,n}$ .

Exact FrontierDelta

Prior state unknown→proved

Scope and record

Occurred: Sep 23, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: Zarankiewicz's crossing-number conjecture (Turan's brick factory problem) · Zarankiewicz crossing-number conjecture

Confidence: Not scored

Registry verification: lean checked · announcement · candidate

Open the source record ↗

Attribution

VibeMathed
registry · event recorded by

Unreleased internal OpenAI model
model · ai model contributor · OpenAI

Evidence vector

SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review

Named evidence

VibeMathed record: Zarankiewicz's crossing-number conjecture (Turan's brick factory problem)

specialist registry · passed

No independent mathematician has checked this yet. Theorem 1.1 was read against the posed problem: $\mathrm{cr}(K_{m,n})=d_md_n$ with $d_r=\lfloor r/2\rfloor\lfloor(r-1)/2\rfloor$ for all positive $m,n$ , ordinary point count, unrestricted drawings. formalization.yaml lists OAI.Zarankiewicz.mainTarget_proof (comparator BipartiteCrossing) as the main result. Its statement MainTarget says that for all $m,n>0$ some admissible drawing of $K_{m,n}$ has exactly $d_md_n$ crossing points and every admissible drawing has at least that many, with the same admissibility conditions as the complete-graph file (simple continuous arcs, finitely many proper crossings, no triple points). That is the headline claim. Permitted axioms are propext, Quot.sound and Classical.choice. Not rebuilt here. Hebbar and Mangam (2018, Int. J. Eng. Technol.) published earlier claims of both formulas, which the manuscripts cite and treat as upper-bound constructions only.