Combinatorics · Oct 6, 2026

Power savings for planar halving lines and k-sets

Improves the planar halving-line bound to

O(n^{4/3-\varepsilon})

for sets with no three collinear and an absolute ε > 0. More generally, an n-point set with no three collinear has

O(n(k+1)^{1/3-\varepsilon_0})

strictly separable k-subsets for

1\le k\le n/2

, with an absolute

\varepsilon_0\gt 0

. The constants and positive exponents are nonquantitative.

Contributors

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