General Position for Planar Line Arrangements and $HD_2(p,3)$

For every $\delta > 0$ and infinitely many $n$ there is a set of $n$ lines in the plane with no intersecting quadruple such that every subset of size at least $n^{4/5+\delta}$ contains three concurrent lines. This improves the bound for a dual form of a theorem of Balogh and Solymosi, and yields an improved lower bound for the Hadwiger-Debrunner number $HD_2(p,3)$ .

Result: Proved(see note)
Status: Partial result
AI contribution: AI-assisted
Method: Construction
Field: Discrete geometry
Posed by: Hugo Hadwiger, Hans Debrunner
Year posed: 1957
Years open: 69y
Solved: 2026-07-28
Model: ChatGPT
Vendor: OpenAI
Collaborators: Oliver Roche-Newton
Verification: Unreviewed
Publication: Preprint
Significance: 15 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

improved bounds; the exact Hadwiger-Debrunner numbers remain open

What the AI did

The AI disclaimer says the work was carried out in collaboration with ChatGPT, while the paper is human-written and the author takes full responsibility for its contents. No individual step is attributed, so the lowest tier applies.

Verification

Single-author arXiv preprint; not yet peer-reviewed.

Source

arXiv:2607.25742 - A general-position problem for planar line arrangements

General Position for Planar Line Arrangements and HD2(p,3)HD_2(p,3)HD2​(p,3)

What was actually shown

What the AI did

Verification

Source

Discussion

General Position for Planar Line Arrangements and $HD_2(p,3)$