Breakthrough Tracker record
Two explicit 64-rectangle counterexamples to Wegner’s packing–piercing conjecture
Two current manuscripts give explicit triangle-free families of 64 axis-parallel rectangles with packing number ν = 16 and piercing number τ at least 32, contradicting Wegner’s bound τ ≤ 2ν − 1. Both also construct recursive families whose packing–piercing ratios and standard clique-LP gaps approach 5/2.
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- Stable ID
math-wegner-rectangle-packing-piercing-counterexample-2026- Revision
math-wegner-rectangle-packing-piercing-counterexample-2026.v1- Field
- Mathematics · Discrete geometry and combinatorial optimization
- Evidence
- Tier 1 · Peer reviewed: No
- Record state
- Provisional · Two explicit preprint constructions; peer review pending
- Last checked
AI role
Bal discloses using OpenAI Codex to format parts of the text and generate figures, followed by author review; no substantive AI role in the mathematical construction is disclosed. The Ajwani et al. manuscript discloses no substantive AI role.
Record details
- Problem or result
- Wegner’s 1965 conjecture that every finite family of axis-parallel rectangles satisfies τ ≤ 2ν − 1
- Authors
- Deepak Ajwani, Rishikesh Gajjala, Rajiv Raman and Saurabh Ray; independently, Aranya Kumar Bal
- Institutions
- University College Dublin, NYU Abu Dhabi, IIIT Delhi and Indian Statistical Institute Kolkata
- Result date
- First preprint submitted June 16, 2026; current companion preprints revised July 9 and July 14, 2026
Why it matters
The conjecture had guided geometric transversal theory since 1965. Besides the compact finite counterexample, the recursive constructions raise the known lower bound for the standard LP relaxation used in maximum-independent-set-of-rectangles algorithms.
Limits
Both sources are revised preprints, not peer-reviewed publications, and no independent specialist correctness assessment was located. Their bibliographic narratives are not yet synchronized: Bal describes an earlier, much larger Ajwani–Gajjala–Raman–Ray construction, while the current Ajwani et al. version now states the same 64-rectangle scale and a 5/2 − ε family. The mathematical claims and priority history should therefore be checked again at formal publication.
Sources
- Primary: Ajwani, Gajjala, Raman and Ray, Counterexamples to Wegner’s Conjecture for Rectangles, arXiv:2606.17854v2
- Primary: Bal, A 64-Rectangle Counterexample to Wegner’s Conjecture and LP Gaps up to 5/2, arXiv:2607.11318v2
- Independent: Caoduro, Cslovjecsek, Pilipczuk and Węgrzycki, prior sharp segment-case context, Journal of Computational Geometry (2023)
Correction and revision history
- 2026-07-23 — Added provisionally after full-text review of both current preprints confirmed the same compact 64-rectangle witness scale and stronger recursive 5/2-limit claims; the absent peer review and inconsistent version/priority narrative are disclosed.
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