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New Bounds for Double Covers of the Discrete Box

For A={0,1,2}dA=\{0,1,2\}^d, write f(d)f(d) for the fewest proper sub-boxes covering every point exactly twice. Leader, Miličević and Tan asked whether f(d)≥2df(d)\ge 2^d for all dd, as Question 4.1 of the PatternBoost paper. The paper gives new bounds on f(d)f(d).

Result
Proved(see note)
Status
Partial result
AI contribution
AI co-developed
Method
Construction
Field
Extremal Combinatorics, Covering Problems
Posed by
Leader, Miličević and Tan
Year posed
2017
Years open
9y
Solved
2026-07-10
Model
Claude
Vendor
Anthropic
Collaborators
Patrick White
Verification
Lean-checked, statement unaudited
Publication
Preprint
Significance
12 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Improved bounds rather than a settled question: the asked-for inequality is not established in general.

What the AI did

The author credits large language models used as reasoning engines with producing several of the key ideas, naming the modular refinement of Section 3, the crossing lemma of Section 5 and the constructions of Section 7.

Verification

Lean formalization available. No independent check. The disclosure is unusually specific about which sections the model produced. Preprint, not refereed.

Sources

Changelog3 changes
  • Matarisvanset More links to “problem-record: Leader et al 2017, Question 4: Problem statement | https://www.sciencedire…”, also Year posed, Verification note, Verification, Significance note
  • Rasmus Lindahlchanged Short name from “Double covers of {0,1,2}^d” to “Double covers of ${0,1,2}^d$”
  • Rasmus Lindahlchanged Short name from “Double covers of $\{0,1,2\}^d$” to “Double covers of {0,1,2}^d”

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