Pach's Tangency Conjecture: Improved Bounds

Pach conjectured that $n$ Jordan arcs, pairwise crossing exactly once with no triple points, have $O(n)$ tangent pairs. The best known bound stood at $O(n^{7/4})$; the paper improves it to $O(n^{3/2})$ (and $O(n^{5/3})$ in the at-most-one-crossing relaxation), plus a tight $\Theta(n^{4/3})$ for a grounded x-monotone variant.

Result

Proved(see note)

Status

Partial result

AI contribution

AI-assisted

Method

Argument

Field

Combinatorial geometry

Posed by

János Pach

Year posed

—

Years open

—

Solved

2026-03-12

Model

Gemini

Vendor

Google DeepMind

Collaborators

Eyal Ackerman, Balázs Keszegh

Verification

Unreviewed

Publication

Preprint

Significance

15 / 100*

Disclosed cost

—

Wikipedia

No dedicated article