Uniform witness conjecture

The uniform witness conjecture, proposed by Chao, Xu, Yip, and Zhang in 2025, predicts the bound $|\mathcal{F}|\leq\binom{n-1}{d}$ for the relevant witness families. A counterexample would disprove it.

Known results

• Chao, Xu, Yip, and Zhang (2025): proved the cases $s=d$ and $s=1$ for sufficiently large $n$.
• Chao, Xu, and Zakharov: proved $s\leq d/2$ for sufficiently large $n$.
• The remaining thin question is whether $\operatorname{W}_{2s-1,s}(n)>\binom{n-1}{2s-1}$ can hold for fixed $s\geq2$ and infinitely many $n$.

June 2026 disproof claim and September 2026 formalization

Zixiang Xu’s June 2026 preprint claims a construction exceeding $\binom{n-1}{d}$ for broad middle ranges of $d$ and $s$, which would refute the conjecture. A September 17, 2026 record additionally reports that the construction has been encoded and proved in Lean, but supplies no independent inspection.

Current status (as of September 2026): The conjecture is claimed to be refuted, while the preprint and Lean formalization remain independently unverified.