5 Maths Problems OpenAI Says Its AI Just Solved

Resolves the Harary–Hill conjecture and Turán's brickyard problem in the Zarankiewicz formulation, determining the crossing numbers of every complete and complete bipartite graph. The result proves the optimality of the classical drawings among all plane drawings with continuous edge arcs. (Lean)

Proves that every five-coloring of the Euclidean plane has a monochromatic pair at distance one, with no restriction on the color classes. This advances the Hadwiger–Nelson problem: together with the classical seven-coloring, only six and seven remain possible chromatic numbers of the plane. (Lean)

Proves the diameter form of Nienhuis's predicted three-quarter exponent: a uniformly chosen n-step self-avoiding walk on the honeycomb lattice has diameter n^3/4+o(1). Its local mass and covering numbers have exponent 4/3. These estimates hold at every sufficiently large fixed length, simultaneously across scales, with arbitrarily high polynomial probability. (Lean)

Determines the isoperimetric profile of the unit cubic flat three-torus and classifies every finite-perimeter minimizer: balls, circular tubes around shortest closed geodesics, coordinate slabs, and their complements. The transition volumes are 4π/81 and 1/π, with exactly the adjacent two types minimizing at each transition. (Lean)

Resolves Schramm’s Hausdorff-measure question for chordal SLEκ, 0 < κ < 8. The explicit gauge r^d(log log(1/r))^(2-d)/2, d=1+κ/8, gives almost surely positive finite measure to every trace segment γ([s,t]) with 0 < s < t < ∞, and finite expected measure to the trace in every bounded disk. (Lean)