Balanced Ham Sandwich Line

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For an affine line $\ell\subset\mathbb R^2$, write $H_\ell^+$ and $H_\ell^-$ for its two open half-planes, in either order. Points on $\ell$ are counted in neither half-plane. For a finite set $X$, its imbalance at $\ell$ is \[ b_X(\ell)=\left||X\cap H_\ell^+|-|X\cap H_\ell^-|\right|. \]

Statement recorded in the supplied source.

For every integer $n\geq2$ and every $n$-point set $X\subset\mathbb R^2$, does there exist an affine line $\ell$ with $|X\cap\ell|\geq2$ and $b_X(\ell)\leq100$? No general-position assumption is made, and the line may contain more than two points. [S2]

Short English statement

Does every finite planar point set admit a line through two of its points with at most one hundred points of imbalance between the two sides?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 136 (GREEN-048). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Ben Green, 100 Open Problems, maintained author collection, Problem 75 and its comments, accessed 7 October 2026. https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf.