Ramos conjecture for two hyperplanes

The Ramos conjecture predicts a specific dimension threshold for partitioning measures equally with two affine hyperplanes. Florian Frick’s new paper claims counterexamples in every integer dimension parameter $n\ge 2$, so the conjecture is false if the construction is correct.

Known results

• A 2015 critical review reported rigorously established threshold values including $\Delta(j,2)=\frac{1}{2}(3j+1)$ when $j-1$ is a power of $2$ and $j\ge 5$, while identifying gaps in several earlier claimed proofs.

September 23, 2026 counterexample

On September 23, 2026, Frick’s article Counterexamples to the Ramos conjecture for two hyperplanes claimed that, for every $n\ge 2$, $4n-2$ smooth nondegenerate Gaussian measures in $\mathbb{R}^{6n-3}$ have no common equipartition, giving $\Delta(4n-2,2)\ge 6n-2$. Combined with upper bounds, it claims $\Delta(2^s-2,2)=3\cdot2^{s-1}-2$ for $s\ge 3$, including $\Delta(6,2)=10$, and analogous perpendicular-hyperplane results.

Current status (as of September 2026): The conjecture is claimed false for the Gaussian two-hyperplane formulation, with several exact threshold values claimed, but the new paper’s conclusions remain unverified.