Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.

Updated Oct 8

Discrete and Convex Geometry

ackerman_2008_there_are_not_too_many_magic_configurations/: Proves Murty's conjecture that a magic configuration of points is in general position, has all its points or all but one collinear, or is the failed Fano configuration.

adiceam_2021_cut_project_quasicrystals_lattices_dense_forests/: Shows single cut-and-project sets are never dense forests, while finite unions of such sets or of translated lattices can be uniformly discrete dense forests.

aggarwal_2026_computer_aided_discovery_extremal_unit_distance/: Develops and compares approximation-based and number-field lattice search methods for finding dense unit-distance graphs in Euclidean spaces and on spheres.

aichholzer_2019_triangles_colored_euclidean_plane/: Classifies triangles by how many plane colors avoid a monochromatic copy: six suffice for almost all triangles and three for near-equilateral ones.

alexeev_2026_short_proofs_combinatorics_probability_number_theory/: Gives the authors' answers to five Erdos problems, on ordinary lines, exponential sums, 4-chromatic graphs, sparse Erdos-Turan, and primes n minus a k squared.

alon_1991_economical_coverings_sets_lattice_points/: Proves that the fewest points of the grid {1,...,n}^d whose connecting lines cover the grid lie between constant multiples of n^{d(d-1)/(2d-1)} and n^{d(d-1)/(2d-1)} log n; for d = 2 this is o(n), answering the question of Problem 798.

alon_2018_uniformly_discrete_forests_poor_visibility/: Proves there is a planar point set with pairwise distances bounded below whose visibility function 2^{C sqrt(log(1/eps))}/eps = eps^{-1-o(1)} is tight up to the o(1) in the exponent.

alon_2026_remarks_disproof_unit_distance_conjecture/: Gives a human-digested proof that some planar point sets determine a fixed power more than linearly many unit distances, disproving Problems 90 and 92.

ambrus_2020_density_estimates_1_avoiding_sets_via/: Improves the upper bound on the density of a planar measurable set avoiding unit distances to 0.25442 using triple-order correlations.

ambrus_2023_density_planar_sets_avoiding_unit_distances/: Proves Erdos's conjecture that a measurable planar set avoiding unit distances has density below 1/4, with the explicit bound 0.2470.

anon_2026_integral_points_norm_one_tori_unit_distance_exponent/: A 13-page manuscript with no author line stating, with a proof via an unramified class field tower, u(n) >= n^(1 + c0 log log log n / log log n) along a sequence.

arman_2017_equally_spaced_collinear_points_euclidean_ramsey/: Shows that for every k at least four, any red-blue coloring of k-dimensional Euclidean space yields a red unit pair or k+3 equally spaced blue collinear points.

arman_2018_result_asymmetric_euclidean_ramsey_theory/: Proves that any red-blue coloring of three-dimensional space contains two red points at distance one or six unit-spaced blue collinear points.

axenovich_2025_ramsey_problems_graphs_euclidean_spaces_cartesian/: Introduces a graph-indexed chromatic number of Euclidean space and determines it for forests and long cycles in terms of the usual chromatic number.

baek_2024_note_erdos_conjecture_about_square_packing/: Proves the Erdős square-packing conjecture in the axis-parallel case, determining every value of the maximum total side length function g.

ball_2009_punctured_combinatorial_nullstellensatze/: Extends grid vanishing to multiplicities and punctures, with geometric covering applications.

ball_2011_erratum_punctured_combinatorial_nullstellensatze/: Removes false coordinate upper bounds in Corollary 4.2 and supplies a counterexample.

balogh_2018_number_points_general_position_plane/: Constructs planar point sets with no four collinear in which every subset of size n^{5/6+o(1)} contains three collinear points.

beeson_2026_solution_erdos_problem_633/: Classifies exactly which triangles can be cut into a non-square number of congruent triangles, settling Erdős problem 633.

behague_2025_nearly_all_known_euclidean_ramsey_sets_subsoluble/: Behague's soluble-orbit enclosures for coordinate-permutation families, finite regular polytopes and symmetric isosceles trapezia.

bellitto_2021_density_sets_euclidean_plane_avoiding_distance/: Proves planar measurable sets avoiding distance one have upper density at most 0.25647 and the plane's fractional chromatic number is at least 3.8991.

berdysheva_2026_duality_delsarte_s_extremal_problem_locally/: Proves strong duality for a generalized Delsarte extremal problem on locally compact abelian groups, unifying finite-group and Euclidean cases.

bezdek_2016_packing_convex_bodies_cylinders/: Proves packing analogs of Bang-type plank and cylinder inequalities, bounding the total cross-sectional volume of cylinders packed in a convex body.

bialostocki_2006_minimum_sets_forcing_monochromatic_triangles/: Shows a planar set forcing a monochromatic copy of a fixed triangle under every 2-coloring needs at least seven points, and gives a seven-point example.

bikeev_2025_isomorphisms_unit_distance_graphs_layers/: Shows Euclidean strips, and l_p layers R^n x [0,epsilon]^m with n >= 2 and 1 < p < infinity, of different widths have non-isomorphic unit distance graphs, and that for n >= 2 the graph of R^n x [0,epsilon] has only isometric automorphisms.

bishnoi_clark_potukuchi_schmitt_2018_polynomial_zeros_finite_grid/: Generalizes the Alon–Füredi nonzero-grid bound to rings with coordinate degree data and applies it to hyperplane covers and blocking sets in finite geometries.

burr_1974_orchard_problem/: Surveys and advances the orchard problem, giving cubic-curve constructions and combinatorial upper bounds for the maximum number of 3-point lines.

cambie_kalviainen_2026_small_step_walk/: Constructs an infinite integer walk in three dimensions with at most sixteen allowed steps and no collinear triple, disproving Problem 193.

clemen_2025_number_regular_simplices_higher_dimensions/: Determines the asymptotic maximum number of regular simplices spanned by n points in R^d, proving a conjecture of Erdos in stronger form.

clisby_2007_self_avoiding_walk_enumeration_via_lace/: Introduces a lace-expansion enumeration method and a two-step algorithm that extend self-avoiding walk and polygon series in all dimensions three and above.

cohen_2023_new_upper_bound_heilbronn_triangle_problem/: Shows that for large n any n points in the unit square contain a triangle of area at most n^(-8/7-1/2000), beating the 1981 Komlos-Pintz-Szemeredi bound.

cohen_2024_clustering_typical_unit_distance_avoiding_sets/: Proves that near-extremal periodic planar sets with few pairs at distance one, and typical discretized sets avoiding distance one, have more than the expected number of pairs at distance 1.96.

cohen_2024_lower_bounds_incidences/: Proves incidence lower bounds for points with tubes, yielding a Heilbronn triangle bound of n^(-7/6+o(1)) for arbitrary point sets.

conlon_2019_lines_euclidean_ramsey_theory/: Reconstructs the exponential Ramsey bounds, periodic coloring and converse, with explicit boundary repairs and three identified source versions.

conlon_2023_more_lines_euclidean_ramsey_theory/: Shows one fixed m admits colorings of every Euclidean space with no red copy of l_3 and no blue copy of l_m, l_k being k collinear points at unit spacing.

conlon_2026_non_spherical_sets_versus_lines_euclidean/: Proves every finite non-spherical set has a length m with colorings of all Euclidean spaces avoiding a red copy of it and m blue collinear points at consecutive distance one.

cranston_2015_fractional_chromatic_number_plane/: Raises the lower bound for the fractional chromatic number of the plane from about 3.5556 to 76/21, roughly 3.6190.

csizmadia_1998_independence_number_minimum_distance_graphs/: Proves that any n points in the plane with minimum distance one contain at least 9n/35 points no two of which are at unit distance.

currier_2024_any_two_coloring_plane_contains_monochromatic/: Proves every two-coloring of the plane contains a monochromatic congruent copy of any three-term arithmetic progression.

currier_2025_more_pointsets_many_rich_lines/: Gives new sharpness constructions for the Szemeredi-Trotter theorem from generalized arithmetic progressions, replacing number theory with incidence geometry.

currier_2026_avoiding_short_progressions_euclidean_ramsey_theory/: Builds colorings of Euclidean space avoiding red and blue short progressions, showing no dimension arrows several small pairs of collinear configurations.

currier_2026_improved_bounds_lines_1_separated_sets/: Gives lattice colorings with no red unit pair or blue 6330-term planar unit progression, and sharper general bounds using local packing.

danzer_1962_zwei_probleme_konvexer_korper_erdos_klee/: Proves that at most two to the n points of Euclidean n-space, not all in a hyperplane, can avoid obtuse triangles, that the same bound governs antipodal sets and touching translates of a convex body, and that the extremal cases are parallelotopes.

decorte_2020_complete_positivity_distance_avoiding_sets/: Characterizes maximum-density distance-avoiding sets exactly as optima of a convex program over completely positive functions, and improves many upper bounds.

ducz_2026_unit_distance_graph_plane_independence_ratio/: Proves that some finite planar unit-distance graph has independence ratio below 1/4, answering a question of Erdős negatively.

duminilcopin_2013_self_avoiding_walk_is_sub_ballistic/: Proves that self-avoiding walk on the integer lattice is sub-ballistic in every dimension at least two.

dvoretzky_1956_covering_circle_randomly_placed_arcs/: Shows that divergence of the arc lengths does not force random arcs to cover the whole circle, gives a sufficient rate of divergence, and poses the covering criterion as an open question.

eberhard_2012_almost_all_sets_d_2_points_d_1_sphere_are_not_subtransitive/: Proves that almost every set of d+2 points on the sphere S^{d-1} is not affinely subtransitive, so spherical configurations need not embed in finite transitive sets.

emmerich_2026_optimizing_explicit_unit_distance_lower_bound_certificates/: Re-optimizes the finite certificate in Sawin's explicit unit-distance bound with Sawin's prime set T, reporting delta = 0.0152616... and u(n) > n^1.0152.

engel_2025_diverse_beam_search_find_densest_known/: Presents a diverse beam search on the Moser lattice that recovers all known maximally dense planar unit-distance graphs and runs the search up to 100 vertices.

erdos_1935_combinatorial_problem_geometry/: Proves that any sufficiently large planar point set in general position contains n points in convex position, with two quantitative proofs.

erdos_1960_extremum_problems_elementary_geometry/: Constructs planar sets with no convex n-gon, giving the exponential lower bound for the convex polygon problem, and bounds the largest forced angle.

erdos_1973_euclidean_ramsey_theorems/: Develops spherical obstructions, field colorings, geometric examples and product proofs, with exact source corrections and external limits.

erdos_1975_euclidean_ramsey_theorems_ii/: Compiles the finite asymmetric and density proof chains, with both source scans, explicit corrections and separate limits for later sections.

erdos_1975_euclidean_ramsey_theorems_iii/: Studies which triangles are forced monochromatically by every two-coloring of the plane, proving a transfer theorem and many families of Ramsey triangles.

erdos_1975_extremal_problems_geometry/: Bounds the maximum number of isosceles, equilateral, congruent and similar triangles determined by n points in Euclidean space.

erdos_1975_problems_elementary_geometry/: Poses problems on how many unit circles and how many distinct circle radii are determined by triples of n planar points.

erdos_1978_more_problems_elementary_geometry/: Claims a bound on the points forcing k whose triples give circles of distinct radii, by an argument later found incomplete, and bounds the least number of convex subsets of n planar points with no three collinear.

erdos_1978_set_theoretic/: A survey of point-set problems, proving that every infinite set in k-space has an equally large subset with all distances distinct.

erdos_1981_applications_graph_theory_combinatorial_methods_number/: A survey of combinatorial and graph-theoretic problems in geometry and number theory, with short proofs and bounds for several of them.

erdos_1982_my_favourite_problems_which_recently_have/: Survey in which Erdős reviews his conjectures in geometry, number theory, combinatorics and analysis that had recently been settled, with references.

erdos_1984_research_problems/: Erdős's problem note on lines determined by n plane points: the conjecture f_k(n) = o(n^2) for k-point lines, the ordinary r-tuple threshold k(n; r, k), large subsets with fewer points on a line, and Beck's theorem that at most n - k points on a line forces ckn lines.

erdos_1987_combinatorial_metric_problems_geometry/: A problem paper on distances among plane points, rich lines, rational distance sets, convex polygons and maximal families of disjoint unit segments.

erdos_1988_solution_problem_grunbaum/: Characterizes, for large n, exactly which integers arise as the number of connecting lines determined by n points in the plane.

escudero_2016_gallai_triangles_configurations_lines_projective_plane/: Answers an Erdős question negatively by exhibiting, for every number of lines above three, arrangements with no Gallai triangle.

exoo_2018_chromatic_number_plane_is_at_least/: Gives an independent proof that the chromatic number of the plane is at least 5, using explicit finite unit-distance configurations.

exoo_2018_hadwiger_nelson_problem_two_forbidden_distances/: Constructs finite plane graphs proving at least five colors are needed when two distances rather than one are forbidden, for several explicit distance ratios.

filho_2019_counterexample_conjecture_larman_rogers_sets_avoiding/: Disproves the 1972 Larman-Rogers conjecture with distance-one-avoiding subsets of the unit ball of volume above two to the minus n of the ball.

fiscus_2024_new_class_geometrically_defined_hypergraphs_arising/: Builds geometrically defined hypergraphs of arbitrarily large edge size with the chromatic number of the Euclidean unit-distance graph, whose proper colorings with that many colors coincide with the graph's.

frankl_1986_all_triangles_are_ramsey/: Proves that for every triangle and every number of colors, all colorings of a high-dimensional Euclidean space contain a monochromatic congruent copy.

frankl_1990_partition_property_simplices_euclidean_space/: Reconstructs the exponential simplex proof through products, approximate grids and exact residual distances, with the hyper-Ramsey subset limit explicit.

frankl_2004_strong_ramsey_properties_simplices/: Frankl and Rödl prove exponential density and color forcing for simplices on spheres arbitrarily close to their intrinsic circumspheres.

fuhrer_2025_progressions_euclidean_ramsey_theory/: Gives an explicit spherical two-coloring of Euclidean space with no red 3-term unit progression and no blue 1177-term unit progression.

furedi_1991_maximal_independent_subsets_steiner_systems_planar_sets/: Shows that every planar set of n points with no four collinear contains more than c sqrt(n log n) points no three of which are collinear, and that the guaranteed size is o(n); also bounds the Erdős-Hajnal set-mapping function g(n) between constant multiples of sqrt n.

ge_et_al_2026_all_simplices_exhibit_canonical_ramsey_property/: Proves that every nondegenerate simplex is canonically Ramsey: in large dimension every r-coloring of Euclidean space contains a monochromatic or a rainbow congruent copy, uniformly in r.

gerver_1979_certain_sequences_lattice_points/: Gives an effective bound forcing K collinear points in long S-walks in Z^2, and shows that in three dimensions an infinite S-walk can have a bounded number of collinear points.

globus_2019_small_unit_distance_graphs_plane/: Classifies unit-distance graphs on at most nine vertices by exhibiting the complete list of 74 minimal forbidden graphs.

graham_1983_euclidean_ramsey_theorems_n_sphere/: Gives a necessary condition for a finite spherical configuration to be sphere-Ramsey in terms of its linear dependences and proves that rectangular bricks of small diagonal are sphere-Ramsey.

graham_2017_euclidean_ramsey_theory/: Handbook chapter stating the plane triangle conjectures of Euclidean Ramsey theory and cataloging the configurations known to be Ramsey.

green_2013_sets_defining_few_ordinary_lines/: Proves the Dirac-Motzkin conjecture and the orchard-planting problem for large point sets, via a structure theorem placing such sets on cubic curves.

grey_2018_chromatic_number_plane_is_at_least/: Constructs finite unit-distance graphs in the plane with no proper 4-coloring, raising the Hadwiger-Nelson lower bound to five colors.

grey_2023_lower_bounds_order_k_chromatic_unit/: Refines lower bounds on the minimum number of vertices and edges of k-chromatic unit-distance graphs in the plane for k equal to five and six; for k equal to seven its vertex bound stays below the best known.

grinsztajn_2026_borsuk_dimension_63_claim/: Unpublished May 2026 proof note claiming a 321-point counterexample in dimension 63, with GPT-5.5 Pro assistance and unreviewed finite checks.

grytczuk_2016_fractional_j_fold_colouring_plane/: Proves at least five colors are needed for plane graphs with distances in a short interval, and gives fractional and j-fold colorings of such graphs.

guruswami_li_2025_density_frankl_rodl_sphere/: Proves density versions of the Frankl–Rödl theorem on the sphere: measurable subsets of S^{n-1} of density sigma, large compared with a negative power of n, contain prescribed simplices with probability polynomial in sigma.

hara_1991_critical_behaviour_self_avoiding_walk_five_more_dimensions/: Announces that in five or more dimensions the self-avoiding walk has purely exponential growth, mean-square displacement linear in the number of steps, and a Brownian scaling limit, with the proofs deferred to the companion papers.

heule_2018_computing_small_unit_distance_graphs_chromatic/: Uses clausal proof minimization to shrink unit-distance graphs of chromatic number 5 down to 553 vertices, far below the previous 1581.

heule_2019_trimming_graphs_using_clausal_proof_optimization/: Presents a proof-optimization method for small unsatisfiable cores and uses it to reduce the smallest 5-chromatic unit-distance graph to 529 vertices.

heule_2024_happy_ending_empty_hexagon_30_points/: Proves by SAT solving that every set of 30 points in general position in the plane contains an empty hexagon, so h(6) = 30.

holmsen_2020_two_extensions_erdos_szekeres_problem/: Extends Suk's Erdos-Szekeres bound to pseudoline arrangements and improves its error term to 2^{n+O(sqrt(n log n))}.

horton_1983_sets_no_empty_convex_7_gons/: Constructs arbitrarily large planar point sets containing no empty convex heptagon, so Erdos's function g(n) does not exist for n at least 7.

hubert_2023_optimization_trigonometric_polynomials_crystallographic_symmetry_spectral/: Minimizes Weyl-invariant trigonometric polynomials via generalized Chebyshev bases and computes spectral chromatic bounds for symmetric set avoiding graphs.

ivan_2026_block_sizes_block_sets_conjecture/: Ivan, Leader, and Walters prove unbounded required block sizes across templates and an optimal degree-two theorem for 123, with source corrections.

ivan_2026_generalised_prisms_euclidean_ramsey_theory/: Ivan, Leader and Walters construct transitive and soluble generalized-prism enclosures and a soluble template extension in their June 2026 preprint.

jackson_2002_sets_meeting_isometric_copies_lattice_exactly_one_point/: Announces a set in the plane meeting every isometric copy of the integer lattice in exactly one point, answering Steinhaus's question in ZFC, with the strengthening that no two of its points are at a distance whose square is an integer.

jacobsen_2016_growth_constant_square_lattice_self_avoiding/: Estimates numerically the square-lattice self-avoiding walk growth constant to fifteen digits, an estimate its authors read as ruling out the long-standing algebraic conjecture for its value.

jelinek_2009_monochromatic_triangles_two_colored_plane/: Every closed-open two-coloring of the plane contains a monochromatic copy of any triangle, and polygonal colorings contain every non-equilateral triangle.

jenrich_2014_two_distance_borsuk_counterexample/: Solo arXiv v6 manuscript with a computational construction in dimension 64 and a 320-point near-counterexample in dimension 63.

jenrich_brouwer_2014_borsuk_counterexample/: Published construction of a 352-point two-distance set in dimension 64 requiring at least 71 parts of smaller diameter.

ji_2026_borsuk_dimension_63_claim/: Retained arXiv v1 claiming a dimension-63 counterexample generated with GPT-5.6 Sol; current v2 was withdrawn after earlier postings were found.

kahn_kalai_1993_borsuk_counterexample/: Published disproof of Borsuk's conjecture, with an eventual exponential lower bound in the square root of the dimension.

kanellopoulos_karamanlis_2019_hales_jewett_type_property_finite_solvable_groups/: Proves a fixed-degree Hales–Jewett-type variable-word property for actions of finite solvable groups, the property the Leader–Russell–Walters approach needs for subtransitive sets to be Euclidean Ramsey.

karamanlis_2022_simplices_regular_polygonal_tori/: Karamanlis embeds every simplex in a finite regular polygon product and deduces its Ramsey property through a transitive abelian group.

kelly_1958_number_ordinary_lines_determined_points/: Shows that n non-collinear points in the real plane determine at least 3n/7 ordinary lines, and bounds the total number of connecting lines.

kirova_2023_two_colorings_normed_spaces_without_long/: Shows every normed space admits a two-coloring in which all sufficiently long unit arithmetic progressions receive both colors.

koizumi_2025_isosceles_trapezoids_unit_area_vertices_sets/: Every planar measurable set of infinite Lebesgue measure contains vertices of a unit-area isosceles trapezoid, isosceles triangle and right triangle.

kovac_2023_coloring_density_theorems_configurations_given_volume/: Colors the plane in 25 colors with no monochromatic unit-area rectangle, and proves matching positive coloring and density results for boxes and simplices.

kovac_2024_polygons_unit_area_vertices_sets_infinite/: Every measurable planar set of infinite measure contains a unit-area cyclic quadrilateral, but some planar set of infinite measure contains no unit-area convex polygon with congruent sides.

kriz_1991_permutation_groups_euclidean_ramsey_theory/: Kříž proves orbit Ramsey coloring for soluble isometry groups, product and orbit-gluing theorems, and regular polygon and polyhedron consequences.

leader_2011_transitive_sets_cyclic_quadrilaterals/: Proves that almost every cyclic quadrilateral, including an explicit symmetric family, does not embed in any finite transitive set.

leader_2012_transitive_sets_euclidean_ramsey_theory/: Leader, Russell and Walters's transitive-set conjecture, equivalent block formulations, uniform three-letter theorem, and non-subtransitive polygons.

lidbetter_2023_improved_bound_gerver_ramsey_collinearity_problem/: Improves the Gerver-Ramsey bound on collinear points in an infinite S-walk in three dimensions from 5^11+1 to 189.

martinez_2015_points_defining_triangles_distinct_circumradii/: Fills a gap in Erdos's proof and shows O(k^9) points in general position contain k whose triples have distinct circumradii.

matolcsi_2025_fractional_chromatic_number_plane_is_at/: Proves that the fractional chromatic number of the unit-distance graph of the Euclidean plane is at least 4.

mauldin_2013_some_problems_ideas_erdos_analysis_geometry/: Surveys problems of Erdős mixing measure theory, geometry and set theory; its Section 5 records the triangle-of-area-one problem with its conjectured constant and argues that the constant is best possible for unions of at most three convex bodies.

medrano_1996_finite_analogues_euclidean_space/: Shows the eigenvalues of the finite Euclidean distance graphs over a finite field of odd order are Kloosterman sums, bounded so that the graphs with nonzero distance are asymptotically Ramanujan.

mirabi_2026_one_point_extensions_euclidean_ramsey_sets/: Mostafa Mirabi's 2026 diagonal and cyclic-product proofs for one-point extensions of finite Euclidean Ramsey sets.

moody_2000_model_sets_survey/: Survey of the cut-and-project construction of model sets, covering their geometry, arithmetic, harmonic analysis, diffraction and dynamics.

moore_2026_pyramid_ramsey_base/: Kenneth Moore's 2026 color-induction proof that an off-affine-hull one-point extension of a Ramsey set is Ramsey.

mundinger_2025_neural_discovery_mathematics_do_machines_dream/: Recasts plane-coloring problems as differentiable optimization, yielding new six-colorings and new triangle-avoiding coloring bounds.

oostema_2020_coloring_unit_distance_strips_using_sat/: Uses SAT solvers on tiled finite instances to find unit-distance colorings of infinite plane strips, raising the 5-color strip height to 1.70084.

openai_2026_atomic_certificate_triangular_lattice_universal_optimality/: A manuscript of the OpenAI mathematics release claiming the covolume-one triangular lattice has the least lower energy per particle among locally finite planar sets of centered-disk density one, for every smooth nonnegative completely monotone function of squared distance, by sharp Gaussian Fourier minorants from a modulo-36 interpolation whose 20-by-20 finite blocks the manuscript checks by exact interval arithmetic; names no Erdős problem, is background for Problem 991, does not apply to 662.

openai_2026_classification_finite_euclidean_ramsey_configurations/: Manuscript of the OpenAI mathematics release claiming a necessary and sufficient tensor criterion over the coordinate field for a finite set to be Euclidean Ramsey at its original scale (Problem 174), with every nonempty subtransitive set and every set of at most five concyclic points Ramsey.

openai_2026_critical_strip_crossing_mass_honeycomb_lattice/: Proves that at the critical weight (2+sqrt 2)^(-1/2) the total weight of honeycomb self-avoiding paths crossing a strip of N bands is comparable to N^(-1/4) and the first horizontal-displacement moment of returning paths is comparable to N^(3/4), by a Yang-Baxter transfer construction, a Pfaffian arch formula and a Bures-Laguerre comparison; formally verified here, the prose unreviewed; honeycomb background for Problem 529.

openai_2026_euclidean_plane_not_five_colorable/: Claims that every five-coloring of the plane with arbitrary color classes has a monochromatic unit pair, so the chromatic number of the plane is 6 or 7, by a transfer to weak measurable colorings through Haar rigidity and a measure-geometric obstruction to five labels; bears on Problem 508.

openai_2026_finite_angular_cylinder_covers_below_half_area_bound/: Constructs finite covers of a regular tetrahedron by cylinders with compact triangular perpendicular bases whose total area is below half the minimum projection area, by tilting the two-cylinder cover sector by sector, a negative answer to the half-area cylinder-covering question and a counterexample to the directionwise 1-codimensional conjecture; both are formally verified here for that tetrahedron, the extension to every tetrahedron is not, and the prose is unreviewed. Names no Erdős problem.

openai_2026_mass_covering_exponents_fixed_length_honeycomb_walks/: A manuscript of the OpenAI mathematics release claiming diameter $n^{3/4+o(1)}$ and local-mass and covering exponent $4/3$ for uniform $n$ -step honeycomb self-avoiding walks at every large $n$ , by renewal and bridge sewing over companion strip and polygon inputs; a honeycomb diameter analogue of the spatial-extent question in Problem 529.

openai_2026_nine_dimensional_counterexample_borsuk_covering_assertion/: Claims the rank-one projectors on R^4 with the Frobenius metric form a compact set in R^9 of diameter sqrt(2) with no cover by ten smaller-diameter sets, so Borsuk's assertion fails in every dimension at least 9; bears on Problem 505. The R^9 statement is formalized in the release, built and axiom-checked here.

openai_2026_planar_point_sets_many_unit_distances/: Gives the original AI-produced proof branch using an unramified pro-3 tower and many fixed split primes to obtain a fixed power of unit distances.

openai_2026_power_improvement_heilbronn_triangle_lower_bound/: Claims n points in the unit square, for every large n, with every triangle of area at least c_1 n^(-2+eta) for one fixed eta > 0, by random integer points in a box under two congruence conditions, lattice counts of small determinants and a deletion step; a power improvement on Komlos-Pintz-Szemeredi; bears on Problem 507.

openai_2026_renewal_changes_law_critical_honeycomb_walks/: Transfers finite critical bridge estimates from companion manuscripts, through Kesten renewal, exact-length Fourier estimates and insertion constructions, into claimed exponent-3/4 spatial laws for infinite, bridge, thermal and uniform honeycomb self-avoiding walks, the uniform lower law only along a density-one set of lengths; only the existence of the free-energy limit and the finiteness of the bridge masses are formally verified here, not the 3/4 laws; analogue of Problem 529.

openai_2026_sharp_fourier_certificate_planar_circle_packing/: A manuscript of the OpenAI mathematics release claiming the planar case of the Cohn--Elkies sharpness conjecture: a radial Schwartz function whose two-point Fourier bound equals the triangular-lattice packing density; it names no Erdős problem and does not apply to Problems 991 or 662.

openai_2026_triangular_minimality_planar_coulomb_renormalized_energy/: Claims the Sandier-Serfaty conjecture, that the covolume-one triangular lattice minimizes the planar Coulomb renormalized energy over all admissible curl-free fields with unit background, by a Voronoi-sector comparison on square tori whose two scalar inequalities the manuscript checks by an interval-arithmetic computation; with the Betermin-Sandier formula this gives the linear term of the minimal logarithmic energy of n points on the two-sphere, the energy of the configurations Problem 991 concerns.

openai_2026_universal_optimality_triangular_lattice/: A 70-page manuscript of the OpenAI mathematics release claiming that the triangular lattice minimizes every completely monotone energy among planar configurations of centered-disk density one, by sharp Gaussian Fourier minorants and positive mixtures; it does not apply to Problems 991 or 662.

palvolgyi_2026_cyclic_non_ramsey_heptagon/: Claims seven points on a circle of transcendental radius that are not Ramsey, against the conjecture that every finite spherical set is Ramsey, by a derivation-weighted EGMRSS argument; arXiv v1 of 20 September 2026.

palvolgyi_2026_nearcircumsphere_ramsey_theorem_solvable_transitive_configurations/: Claims that every finite spherical set with a solvable transitive isometry group is monochromatically forced on high-dimensional spheres of radius slightly above its circumradius, through a Kneser-shift graph theorem proved by a Z_p-Tucker lemma; arXiv v1 of 11 August 2026.

parts_2020_chromatic_number_plane_is_at_least/: Gives a proof that the chromatic number of the plane is at least 5 which a person can check by hand, without computer assistance.

parts_2020_graph_minimization_focusing_example_5_chromatic/: Introduces a property-preserving graph minimization method and applies it to obtain a 5-chromatic unit-distance graph with 509 vertices and 2442 edges.

parts_2020_what_percent_plane_can_be_properly/: Constructs partial tilings properly covering over 99.9856 percent of the plane with six colors and over 95.99 percent with five colors.

parts_2022_plane_coloring/: Gives a human-verifiable proof that the plane needs exactly 7 colors when every distance in [1, d] is forbidden, for each d with 2 sin(2 pi / 9) (about 1.285575) < d <= sqrt(7)/2.

patel_2025_biggest_open_problem_euclidean_ramsey_theory/: Surveys what is known about which finite point sets are Ramsey in high-dimensional Euclidean space, and the rival characterizing conjectures.

pohoata_2022_convex_polytopes_fewer_points/: Shows the Erdos-Szekeres function in dimension three and above is subexponential, so ES_d(n) = 2^o(n) for all d >= 3.

poonen_rubinstein_1995_number_intersection_points_made_by_diagonals_regular_polygon/: A theorem-indexed source review of the arXiv v3 text.

prosanov_2020_new_proof_larman_rogers_upper_bound/: Gives a new proof, avoiding Butler's theorem, that the chromatic number of n-dimensional Euclidean space is at most (3+o(1))^n, the Larman-Rogers bound.

protasov_2024_optimal_partitions_flat_torus_into_parts/: Determines the least maximal part diameter for partitions of the flat 2-torus into three parts and gives numerical bounds up to 25 parts.

purdy_2009_lines_circles_planes_spheres/: Gives lower bounds on the numbers of planes and spheres determined by n points in space, the sphere bound being tight and tied to the orchard problem, and corrects Elliott's lower bound for circles in the plane.

putterman_2026_infinite_sets_no_line/: Constructs an infinite planar set with dense collinear-free subsets that cannot be split into finitely many such sets.

raigorodskii_2000_chromatic_number_space/: Proves that the chromatic number of n-dimensional Euclidean space is at least (1.239...+o(1))^n, improving Frankl and Wilson's base 1.207.

regev_stephens_davidowitz_2016_reverse_minkowski_theorem/: Develops reverse-Minkowski and stable-lattice Gaussian estimates.

ruhland_2025_no_new_lower_bound_density_planar/: Withdraws the author's earlier claim of a density improvement for planar unit-distance-avoiding sets, reporting that none of the sets of constant diameter it investigates beats Croft's 0.22936 lower bound.

sawin_2026_explicit_lower_bound_unit_distance_problem/: Gives a quantitative CM-field and class-tower construction of arbitrarily large planar point sets with at least a constant times n^1.014114 unit pairs.

shader_1976_all_right_triangles_are_ramsey_e2/: Shows that every right triangle is Ramsey in the two-colored plane, plus two further algebraic families of Ramsey triangles.

shaw_2026_regular_pentagon_canonically_ramsey/: Shaw proves canonical Ramsey theorems for powers of prime polygons and describes divisor obstructions for composite polygon product hosts.

shirandami_2023_dense_forests_constructed_grids/: Characterizes when finite unions of translated lattices are dense forests, and shows that, for almost all rotations, unions of enough rotated lattices have visibility bounds arbitrarily close to the optimal one.

shkredov_2015_problems_euclidean_ramsey_theory/: Gives Bessel-function criteria under which every measurable two-coloring of the plane contains a prescribed monochromatic triangle or collinear triple.

singh_2026_square_packing_conjecture_erdos/: Shows Erdos's conjecture that f(k^2+1)=k for square packings is equivalent to convergence of the series of excesses f(k^2+1)-k.

sokolov_2025_chromatic_number_plane_map_type_colorings/: Shows that 7 colors are needed for polygonal colorings of the plane, and for locally finite map-type colorings with no unit-curvature boundary arcs and no trichromatic vertex of degree above 3.

solymosi_2013_many_collinear_k_tuples/: Constructs n-point planar sets with no k+1 collinear points yet nearly n^2 collinear k-tuples, for every k at least 4.

suk_2017_erdos_szekeres_convex_polygon_problem/: Proves ES(n) = 2^{n+o(n)}, nearly settling the Erdos-Szekeres convex polygon conjecture.

swanepoel_2002_independence_numbers_planar_contact_graphs/: Raises the lower bound for the independence number of a planar minimum-distance graph on n points to 8n/31, and beats n/4 for non-paralleloid convex discs.

szemeredi_1983_extremal_problems_discrete_geometry/: Proves the incidence bound of order n to the two thirds times t to the two thirds for n points and t lines in the plane, and derives from it the bound n squared over k cubed on k-rich lines, a point on linearly many of the determined lines, and an exp of order root n count of line-size sequences.

tolmachev_2025_lower_bounds_density_planar_periodic_sets/: Recasts the maximal density of planar unit-distance-avoiding sets as independent-set search on flat-torus graphs, finding no improvement on Croft's bound.

tsaturian_2017_euclidean_ramsey_result_plane/: Proves that in every red-blue coloring of the plane there is a red unit-distance pair or a blue five-term progression with unit spacing.

tsaturian_2019_problems_extremal_graph_theory_euclidean_ramsey/: A thesis on extremal numbers for unions of color-critical graphs, cycle counts in triangle-free, edge-bounded and random graphs, and two asymmetric Euclidean Ramsey results.

vallentin_2025_conic_optimization_extremal_geometry/: Surveys conic optimization bounds for packing: kissing numbers, angle-avoiding spherical sets, sphere packing and measurable one-avoiding sets.

vinh_2005_chromatic_number_unit_quadrance_graphs_finite/: Bounds the chromatic number of the graph on a finite plane joining points at unit quadrance between about half the square root of q and about q/2.

voronov_2022_constructing_5_chromatic_unit_distance_graphs/: Constructs new 5-chromatic unit distance graphs in the plane, some avoiding the Moser spindle, and on two spheres of specified radii.

voronov_2025_chromatic_number_plane_interval_forbidden_distances/: Proves that any plane coloring forbidding all distances in a short interval around 1 needs at least 7 colors, for every planar norm.

vritsiou_2023_regular_ellipsoids_blaschke_santalo_type_inequality_projections_non_symmetric_convex_bodies/: Extends Pisier's regular ellipsoid estimates to non-symmetric convex bodies.

wormald_1979_chromatic_graph_special_plane_drawing/: Shows that some finite set of points in the plane has a 4-chromatic unit-distance graph of girth 5, answering Erdos's question negatively when cycles of length 3 and 4 are excluded.

zhang_2025_tiling_triangles_angles/: Constructs infinite families of tilings of triangles by congruent triangles having a 2 pi/3 angle, and conjectures that their tile counts are the only possible ones.

zhao_ge_2026_monochromatic_unit_equilateral_triangle_low_dimensional_spheres/: Determines the least sphere dimension in which every red–blue coloring of the sphere of radius 1/sqrt 2 contains a monochromatic unit equilateral triangle: it is 3.

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