Remark: Family 183. Power savings for planar halving lines and k -sets

Improves the planar halving-line bound to $O(n^{4/3-\epsilon })$ for sets with no three collinear and an absolute $\epsilon >0$. More generally, an $n$-point set with no three collinear has $O(n(k+1{)}^{1/3-{\epsilon }_0})$ strictly separable $k$-subsets for $1\le k\le n/2$, with an absolute ${\epsilon }_0>0$. The constants and positive exponents are nonquantitative.

The authors are OpenAI. This companion does not verify the proof. The claim is not a theorem of the Enter series and not a premise of any registered sheet. It does not close O.1. O.1 stays NOT_CLOSED. It is not a de Sitter state and not the QdS external-leg scheme. No quantum coefficient is admitted. No compilation is asserted.