Distances and packing
How points can be spread out, how often a distance can repeat and how tightly shapes can be packed.
5 AI-made results on distances and packing, from 6 papers in the OpenAI math collection, each explained in plain English. 4 have a formal Lean proof listed; 1 is claimed in a paper and not yet machine-checked. They span 3 fields, led by combinatorics (3), differential geometry (1) and theoretical computer science (1). The oldest question here dates back to 1946: The higher-dimensional Erdős distinct-distances conjecture. Rated Landmark: Planar distinct distances and unit-distance bounds.
Theoretical computer science1
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Bin packing and unbounded configuration-LP gaps
Disproves the modified integer round-up conjecture of Scheithauer and Terno: the integral bin-packing optimum can exceed its configuration linear-programming value by an arbitrarily large additive constant. Approximating the optimum within any fixed additive constant is also NP-hard, even when every item exceeds 1/6 and each bin holds at most five items.
Formal proof listed
Combinatorics3
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The higher-dimensional Erdős distinct-distances conjecture
For every fixed d ≥ 3, any n ≥ 2 distinct points in ℝd determine at least distinct distances, with depending only on dimension. This matches the integer-grid order and resolves the higher-dimensional Erdős distinct-distances conjecture with a constant-factor bound.
Claimed, not checked -
Planar distinct distances and unit-distance bounds
Proves the weak pinned Erdős distance conjecture: for every fixed ε > 0, all but points of any n-point planar set determine at least distinct nonzero distances. A complementary theorem bounds the number of unit-distance pairs by for an absolute δ > 0.
Formal proof listed -
Counterexamples to infinite matroid intersection and packing/covering
Disproves the unrestricted infinite matroid intersection and packing/covering conjectures in ZFC, using two self-dual partitional matroids on a countably infinite ground set. The same examples answer Joó’s partitional-matroid question negatively. They are neither finitary nor cofinitary, so Nash-Williams’ original finitary conjecture remains outside the result.
Formal proof listed
Differential geometry1
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Symplectic ball packing in higher dimensions
Resolves the Siegel–Yao conjecture for arbitrary capacities in every dimension . Finitely many closed symplectic balls of capacities embed disjointly into an open ball of capacity R exactly when and for every distinct pair i, j.
Formal proof listed