Wegner's Piercing Conjecture for Rectangles

Wegner conjectured in 1965 that every finite family $\mathcal{R}$ of axis-parallel rectangles satisfies $\tau(\mathcal{R}) \le 2\nu(\mathcal{R}) - 1$ , where $\tau$ is the minimum number of piercing points and $\nu$ the largest pairwise-disjoint subfamily. False, by an explicit triangle-free counterexample.

Result: Disproved
Status: Resolved
AI contribution: AI co-developed
Method: Construction
Field: Discrete geometry
Posed by: Gerd Wegner
Year posed: 1965
Years open: 61y
Solved: 2026-06-16
Model: GPT-5.5 Pro, Codex
Vendor: OpenAI
Collaborators: Deepak Ajwani, Rishikesh Gajjala, Rajiv Raman
Verification: Site-confirmed
Publication: Preprint
Significance: 25 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What the AI did

The authors say the construction of the initial counterexamples relied heavily on trial and error, and that GPT-5.5 Pro was used extensively to search for suitable constructions. Codex drew the figures and drafted portions of the text. They note the correctness of the proofs does not rest on the auxiliary machine verifications.

Verification

The refutation is an explicit finite family of rectangles, so it reduces to a finite piercing computation. arXiv preprint, not peer-reviewed.

Source

arXiv:2606.17854 - Counterexamples to Wegner's Conjecture for Rectangles

Wegner's Piercing Conjecture for Rectangles

What the AI did

Verification

Source

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