Pach's Tangency Conjecture: Improved Bounds
Prior state unknown→proved
Exponent improvements toward Pach's conjecture, which remains open.
SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review
combinatorics / Combinatorial geometry
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.
15Significance / 100
1Frontier events
0Verification tasks
0Recorded attempts
Temporal state
Current frontier
No reconciled state yet.
Append-only history
Frontier timeline
Research memory
Claims and attempts
Scoped claims
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.
Exponent improvements toward Pach's conjecture, which remains open.
Recorded attempts
Evidence graph
Connected research record
No public relationships recorded yet.