{"schema_version":"dataset-v1","id":"erdos-unit-distance-exponent","version":1,"title":"Where the growth exponent lies","description":"Let U(n) be the most unit-distance pairs among n planar points, and θ = lim sup log U(n) / log n its growth exponent. Erdős conjectured θ = 1, that is U(n) = n^(1+o(1)). The 2026 construction proves θ ≥ 1 + δ for some δ > 0; Spencer, Szemerédi and Trotter proved θ ≤ 4/3.","license":"CC0-1.0 (exact values)","problems":["erdos-unit-distances"],"statements":[{"problem":"erdos-unit-distances","statement":"unit-distances-near-linear","role":"Erdős’s conjecture θ = 1, now refuted."},{"problem":"erdos-unit-distances","statement":"unit-distances-positive-exponent","role":"The 2026 construction: θ ≥ 1 + δ for a fixed δ > 0."},{"problem":"erdos-unit-distances","statement":"unit-distances-four-thirds-upper","role":"Spencer–Szemerédi–Trotter: θ ≤ 4/3."},{"problem":"erdos-unit-distances","statement":"unit-distances-erdos-grid","role":"Erdős’s grid: superlinear, but its gain c/log log n vanishes."}],"figure":{"family":"region-plot","title":"Where the growth exponent lies","caption":"The true exponent θ is now known to lie strictly above 1 and at most 4/3. The 2026 gain δ is not explicit, so the open interval is drawn from 1, which is itself excluded. Erdős’s grid gives a gain c/log log n that tends to zero, which is why the conjecture θ = 1 was plausible.","x_label":"exponent θ","y_label":"","x_range":[0.95,1.4],"y_range":[0,1],"y_scale":"linear","x_ticks":[[1,"1"],[1.1,"1.1"],[1.2,"1.2"],[1.3333333333333333,"4/3"]],"layers":[{"key":"open","label":"Open: the true exponent, now known to exceed 1","kind":"polygon","status":"open","statement":"unit-distances-positive-exponent","x":null,"y_max":null,"series":null,"points":[[1,0],[1.3333333333333333,0],[1.3333333333333333,1],[1,1]]},{"key":"conjecture","label":"Target: θ = 1, Erdős’s conjecture, refuted","kind":"line","status":"target","statement":"unit-distances-near-linear","x":1,"y_max":null,"series":null},{"key":"upper","label":"Proved: θ ≤ 4/3 (Spencer–Szemerédi–Trotter)","kind":"line","status":"proved","statement":"unit-distances-four-thirds-upper","x":1.3333333333333333,"y_max":null,"series":null}],"supersedes":null},"provenance":{"method":"No computation: the interval endpoints 1 and 4/3 come from the cited statements. δ is not explicit in the sources, so no lower edge above 1 is drawn.","software":{"python":"3.12.15"},"precision":"Exact rationals, stored as floating point.","generator":"openmath/scripts/build-datasets.py","generated_on":"2026-10-21"},"data":{"series":[{"key":"endpoints","label":"θ = 1 (conjectured) and θ = 4/3 (proved upper bound)","x":[1,1.3333333333333333],"y":[0,0]}],"marks":[]},"sha256":"632e069fb580685211e15c25801550952f0a87755914ba59762ea10b30d7569c"}