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

Updated Oct 8

Problem 188

claims/: The 3 claim pages of Problem 188, one per claimant's result; the problem's standing derives from them.

Statement. What is the smallest $k$ such that $\mathbb{R}^2$ can be red/blue coloured with no pair of red points unit distance apart, and no $k$ -term arithmetic progression of blue points with distance $1$ ?

Formulation. Erdős and Graham asked the question with no restriction on the step of the blue progression [ErGr79, pp. 330–331; ErGr80, pp. 14–15]: how small can a positive integer $M$ be when the plane is split into a set with no two points at distance one and a set with no arithmetic progression of length $M$ (the two passages). That question has the answer that no finite $M$ exists. As the site's commentary records, Alon observed that on the integer points of a line a coloring with no red unit pair makes the blue points and their translate by $1$ cover all of them, so van der Waerden's theorem gives arbitrarily long blue progressions (the complete argument). The commentary judges that Erdős and Graham most likely intended the unit step, though their papers do not write it, and the site's Statement asks the question with it; that question sets the standing.

In the notation of this page, the Statement reads as follows. What is the smallest positive integer $K_*$ for which the whole plane can be red-blue colored with no red pair at distance $1$ and no blue progression

x,x+v,\ldots,x+(K_*-1)v,\qquad |v|=1?

Both the location $x$ and the direction of the unit vector $v$ are arbitrary. The length counts points, so a five-term progression has four unit gaps. "With distance $1$ " means a unit common difference, as the site's remarks explain.

Status. Open: the site labels the problem OPEN. The public sources compiled as of 2026-09-13 give

6\leq K_*\leq6330.

The lower bound is Tsaturian's refereed theorem (claim page). The upper bound is from the revised Currier–Mody–Xie–Zhang preprint (claim page); the best refereed upper bound is Conlon and Fox's $10^{10}$ (claim page). The site's commentary, last edited 14 October 2025, does not cite the Currier–Mody–Xie–Zhang bound. These are the compiled source-backed bounds, not the outcome of an exhaustive literature census.

Source. erdosproblems.com/188, accessed 2026-09-05, together with the discussion and empty proof-claim thread. Cite as: T. F. Bloom, Erdős Problem #188, https://www.erdosproblems.com/188, accessed 2026-09-05. The problem page was last edited October 14, 2025. Its caution about the original formulation matters: the intended blue progressions have unit step. The site attributes to Alon the observation that without this restriction van der Waerden's theorem prevents such a coloring for any finite length. The original wording is in both the 1979 paper and 1980 monograph. The complete van der Waerden and translation argument explains why the omitted step restriction is necessary.

References.

[ErGr79] P. Erdős and R. L. Graham, Old and new problems and results in combinatorial number theory: van der Waerden's theorem and related topics, L'Enseignement Mathématique (2) 25 (1979), 325–344; the question is on pp. 330–331.
[ErGr80] P. Erdős and R. L. Graham, Old and new problems and results in combinatorial number theory, Monographies de L'Enseignement Mathématique 28 (1980); the same question is on pp. 14–15. See the source record.
Erdős, Graham, Montgomery, Rothschild, Spencer, and Straus, Euclidean Ramsey theorems. II (1975), 529–557.
Tsaturian, A Euclidean Ramsey Result in the Plane, Electronic Journal of Combinatorics 24(4) (2017), P4.35, doi:10.37236/7148.
Currier, Mody, Xie, and Zhang, Improved bounds for lines and $1$ -separated sets in Euclidean Ramsey theory, arXiv:2606.17194v2, August 31, 2026.

Formalization. The pinned public statement has unfinished proofs. See the formalization account below for its definition history.

Current assessment

Tsaturian's Theorem 1 proves that every plane coloring has a red unit pair or five blue points in a unit progression. Thus no avoiding coloring exists for length five, and $K_*\geq6$ . The complete published proof uses forced lattice configurations, a classification into two periodic patterns, and a final off-lattice choice of a unit-distance pair on a radius-five circle.
Currier–Mody–Xie–Zhang's Theorem 1.2, arXiv:2606.17194v2, gives a periodic plane coloring avoiding a red unit pair and every blue $6330$ -term unit progression. Its probability proof counts the admissible cell placements of unit progressions in every direction, assigning each cell wall to one neighboring cell, and settles its two numerical steps, the exponent $0.01557$ of Lemma 3.2 and the threshold $6330$ , by a short calculation that it does not display; the library's reconstruction makes the boundary conventions explicit, and its evidence script certifies both steps with exact rational bounds. The reconstruction has no independent review.
The same paper's Theorem 1.1 gives a general-dimensional upper bound in terms of separation, diameter and local density. The August 31 revision improves its general exponential base to $6.79$ through spherical codes. It leaves the planar $6330$ endpoint unchanged.

The linked modern and historical source units retain reconstructed proof chains, with the Currier paper's external theorem inputs stated separately. Those source records do not document independent acceptance of the local reconstructions. Their retention does not discharge the outstanding literature-compilation review obligations.

If a coloring avoids a blue progression of length $k$ , it avoids all longer ones by taking a consecutive $k$ -term subset. Thus the set of admissible lengths is upward closed; the upper-bound source proves it is nonempty. The question concerns its least member, not the largest length for which a Ramsey conclusion holds.

Historical and current-source scope

Erdős–Graham–Montgomery–Rothschild–Spencer–Straus established the earlier four-blue-point statement. Their Theorem 1′ has a complete planar proof using two concentric circles. The materially different Theorem 1 uses forced colors on a triangular lattice in three dimensions, relative to the explicitly stated monochromatic-triple theorem from Paper I. The planar result gives the historical bound $K_*\geq5$ , superseded by Tsaturian's $K_*\geq6$ ; it gets no claim page, since it appeared in a proceedings volume (Colloq. Math. Soc. János Bolyai 10) and Tsaturian's refereed theorem contains it. The Conlon–Fox paper proved general upper bounds, including the planar estimate $10^{10}$ quoted by the newer source and recorded on its own claim page. The site's discussion reports a better constant extractable from that earlier proof; that separate optimization is not reconstructed here. The old estimate near $10^7$ in the Erdős–Graham book was given without a proof in the cited discussion.

The site's discussion thread, when read, held three comments: a correction to an arXiv lemma's side length, a Conlon–Fox constant calculation, and a warning about an older formalized statement. The published Tsaturian version corrects the triangle side length; its remaining color slips are documented in the source digest. No proof claim or linked exposition appeared in the checked thread.

The Tsaturian source record keeps its journal and arXiv version-history checks. The Currier source record keeps its primary arXiv record and main-statement checks, and reports bounded searches, not an exhaustive census. The August 31 Currier revision is preserved as the canonical source with its older version retained.

A bounded bibliographic check revisited the cited papers' primary abstract, version-history and publication records and the corresponding author listing. It supplies no source-backed change to the compiled numerical bounds. This check does not establish exhaustive coverage, priority, proof correctness or the exact minimum.

Formalization

The pinned formal-conjectures file contains the intended definition with arbitrary complex direction of norm one. Its least-value answer, main proof, and variants contain sorry; it is not a formal solution. An older definition fixed the horizontal direction. PR 3890, merged April 28, 2026, corrected that defect. The forum warning therefore concerns the older definition, not the current one. The recorded account does not specify the depth of source inspection for these definition comparisons or document an independent statement-fidelity review.

Connections and remaining work

The related Problem 214 asks for a blue unit square under the same red-unit-pair exclusion. Its fixed small configuration requires a different lower-bound argument; the general large-configuration coloring theorem does not settle it. The public lower-bound and upper-bound methods here give two complementary tools: local forced-color propagation and probabilistic periodic cell selection.

OpenAI's preprint The Euclidean plane is not five-colorable (OpenAI Math Release, 23 September 2026, pinned PDF, carded at openai_2026_euclidean_plane_not_five_colorable) proves that every coloring of the plane with five colors, with no regularity assumption, has a monochromatic unit-distance pair, so the chromatic number of the plane is six or seven. That is a result on Problem 508, where it belongs. It claims nothing about this problem and gets no claim page here.

The compiled sources do not determine the exact minimum. The Conlon–Fox source record describes a reconstructed proof chain and an explicitly compilation-supplied boundary repair. This source-record check does not verify that reconstruction or the separate sharper constant reported in the retained catalog discussion. The original four-point arguments are compiled in the linked 1975 source record, and the historical formulation is checked in the linked 1979 paper and 1980 book.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

Linked from (116)

Problem 174 Problem 188 Tsaturian, a red unit pair or a blue five-term unit progression Conlon and Fox, an avoiding coloring for unit progressions of 10^10 terms Currier, Mody, Xie and Zhang, an avoiding coloring for 6330-term unit progressions problems/discrete_geometry/E0188/claims Discrete and Convex Geometry Problem 214 additive_combinatorics/erdos_1979_old_new_problems_results_combinatorial_number The historical plane-coloring question on pages 330–331 Why the blue progression needs a prescribed step analysis/janson_1998_new_versions_suen_correlation_inequality Theorem 2 (p. 3): P(S=0) <= exp(-mu + sum of E(I_iI_j) exp(sum_{k~{i,j}} p_k)) <= exp(-mu + Delta e^{2 delta})Discrete and Convex Geometry discrete_geometry/adiceam_2021_cut_project_quasicrystals_lattices_dense_forests Proposition 2.1 (p. 7): when translates of given lattices have a uniformly discrete union Theorem 1.1 (p. 2): a cut-and-project set is never a dense forest Theorem 1.2 (p. 3): uniformly discrete dense forests from finitely many cut-and-project sets Theorem 1.3 (p. 3): three translated lattices forming a uniformly discrete dense forest Theorem 1.4 (p. 3): almost every union of ns lattices is a dense forest with near-optimal visibility Theorem 5.2 (p. 18): visibility of Peres-type lattice forests from a uniform Diophantine condition discrete_geometry/alon_2018_uniformly_discrete_forests_poor_visibility Theorem 1.1: a uniformly discrete forest with visibility eps^{-1-o(1)}discrete_geometry/arman_2017_equally_spaced_collinear_points_euclidean_ramsey Conjecture 1 (p. 4): one k with E^n not arrowing (l_3, l_k) for every n Theorem 2.1 (p. 2): E^k arrows (l_2, l_{k+3}) for every k >= 4 discrete_geometry/arman_2018_result_asymmetric_euclidean_ramsey_theory Theorem 2.1: E³ → (ℓ₂, ℓ₅)Theorem 3.1: E³ → (ℓ₂, ℓ₆)discrete_geometry/conlon_2019_lines_euclidean_ramsey_theory The constants in the union bound Asymmetric Ramsey and periodic-cell conventions External inputs and historical examples Lemma 2.1: the number of net points Lemma 2.2: local packing Lemma 2.3: Voronoi facets Lemma 2.4: a sparse subset The line counterexample in every dimension Linear sign patterns, including zeros Avoiding both colors for one configuration Periodic selection and independent neighborhoods Theorem 1.2: large separated configurations Theorem 1.3: Szlam’s exponential lower bound Theorem 2.5: the imported polynomial sign bound Theorem 3.1: the general first-red-index transfer Theorem 3.2: a converse from finite compactness The unit-sphere observation discrete_geometry/conlon_2023_more_lines_euclidean_ramsey_theory Conjecture 4.2 (p. 4): every non-spherical X has an m with E^n not arrowing (X, l_m)Theorem 1.1 (p. 1): one m with E^n not arrowing (l_3, l_m) for every n

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