Pith. sign in

REVIEW 3 major objections 5 minor 10 references

Ramsey Theory on the Integer Grid: The "L" Problem

T0 review · 3 major / 5 minor · reviewed 2026-08-08 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read The paper proves that every 3-coloring of a 493 by 493 integer grid contains a monochromatic L shape, improving the known upper bound from 2593.

desk verdict The interval-counting method is new and improves the bound, but the final 493 rests on a computational claim that is not verifiable from the written proof. read the letter →

arxiv 2502.05162 v1 pith:DCK7K4SC submitted 2025-02-07 math.CO

Isaac Mammel , William Smith , Carl Yerger This is my paper · ORCID
classification math.CO MSC 05D10
keywords RamseytheoryintegergridmonochromaticLGolombrulerintervalcountingSATsolver3-coloring
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper takes on a Ramsey-theoretic question: how large must an $n \times n$ integer grid be before every 3-coloring of its points forces a monochromatic L, meaning three same-colored points $(i,j)$, $(i+t,j)$, and $(i+t,j+t)$? It proves that a 493 by 493 grid already guarantees such an L, replacing the previously known upper bound of 2593 with a much smaller number. The argument counts same-colored points on the main diagonal, then shows that points forced into other colors on diagonals below it cannot accumulate without themselves forming an L; the final step uses the fact that forced blue points on such a diagonal must have all pairwise distances distinct, i.e., form a Golomb ruler. The paper also reports SAT-solver searches showing that a 20 by 20 grid with no monochromatic L exists, while a 21 by 21 grid has not yet been found.

What carries the argument

The key mechanism is an interval-counting argument on the main diagonal of the grid. Consecutive red points on the diagonal define intervals; each interval of length $c$ forces a point on subdiagonal $S_{c+1}$ to be blue or green, and too many forced points of one color on the same subdiagonal force a monochromatic L. Later the paper also counts intervals between nonconsecutive red points, and finally imports Golomb rulers, sets of integers in which every pair has a distinct distance, to replace the crude bound $b$ with the distance-uniqueness bound $b_k$ on each subdiagonal.

What would settle it

Try to construct a 3-coloring of a $493\times493$ grid that avoids monochromatic Ls, or find a diagonal in the proof's forced-point construction where two pairs of forced blue points share the same distance yet the grid still has no L; either would break the central claim.

Watch

Extended reading notes

Core claim

The paper's central claim is that $R_3(L) \le 493$, meaning any 3-coloring of a $493\times493$ integer grid contains a monochromatic L. It reaches this by first establishing upper bounds of 1804, 1573, and 772 through increasingly careful interval counting on the main diagonal. The decisive improvement replaces the earlier uniform bound on the number of forced blue points per subdiagonal (a diagonal of the grid parallel to the main diagonal) with $b_k$, the largest order of a Golomb ruler of length $n-k-1$, because forced blue points on a subdiagonal cannot repeat a pairwise distance without forcing a monochromatic L. Applying this refined bound with $c=12$ yields the contradiction at $n=493$.

Load-bearing premise

The 493 bound relies on the assumption that forced blue points on the same diagonal below the main diagonal cannot be the same distance apart without creating a monochromatic L, and this Golomb-ruler property is asserted without a fully detailed proof.

Editorial extensions

If this is right

  • Every 3-coloring of any $n\times n$ grid with $n\ge 493$ must contain a monochromatic L, since a 493 by 493 subgrid already forces one.
  • The interval-counting technique together with Golomb rulers produces the chain of upper bounds 1804, 1573, 772, and 493, a substantial reduction from the earlier 2593 bound.
  • The lower bound remains 21: the paper verifies with SAT solvers that 20 by 20 grids without monochromatic Ls exist, and no 21 by 21 grid has been found.
  • Theorem 8 connects the lower bound to arithmetic-progression Ramsey numbers, giving $R_3(L) \ge 14$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the Golomb-ruler step is valid, the bound 493 is not obviously optimal: using optimal Golomb rulers for every subdiagonal length, or tuning the parameter $c$ in the algorithm, could plausibly lower the threshold further.
  • The same interval-forcing scheme would apply to 4 colors by iterating the forcing chain twice, and the large drop from 2593 to 493 suggests the true value of $R_3(L)$ may be far below the new upper bound.
  • The SAT-solver evidence that 20 by 20 grids exist without monochromatic Ls, while 21 by 21 remains unsolved, hints that the true answer may lie near 21, meaning the current upper bound could still be off by more than an order of magnitude.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper studies the least integer R_3(L) such that every 3-coloring of an n x n integer grid contains a monochromatic L-shaped triple. Building on the Gallai-Witt corollary and the previous upper bound 2593, the authors develop interval-counting arguments on subdiagonals, introduce counts over non-consecutive intervals, and finally use Golomb rulers to claim successive upper bounds of 1804, 1573, 772, and 493. The manuscript also reports SAT-solver experiments for the lower bound and proves a weak lower bound from van der Waerden numbers.

Significance. If the claimed bounds are correct, the reduction of the known upper bound for R_3(L) from 2593 to 493 is a substantial and interesting improvement. The main mechanisms, especially the Golomb-ruler restriction on forced blue points, are elegant and potentially reusable in similar grid Ramsey problems. However, the paper currently ships no machine-checked scripts, Theorem 6 and Algorithm 3 do not expose the decisive arithmetic, and Lemma 1 contains an algebraic error in the definition of b; the numerical results therefore cannot yet be certified from the manuscript as written.

major comments (3)
  1. [§2.1, Lemma 1] The derivation of b is arithmetically incorrect. From C(b,2)/(n-2) > 1 one obtains b^2 - b > 2n - 4 and hence b^2 - b + 1/4 > (8n - 15)/4, so b > sqrt((8n - 15)/4) + 1/2, not b > sqrt((2n - 15)/4) + 1/2. The printed formula gives b approximately sqrt(n/2), while the later text requires b approximately sqrt(2n); indeed Section 2.2 states b = 61 for n = 1803, which is only consistent with the corrected formula. Since b is used in Theorems 3, 4, and 5 and in Algorithms 1 and 2, this error must be fixed and the numerical thresholds re-verified.
  2. [§2.3, Theorem 6 and Algorithm 3] The proof of Theorem 6 is not verifiable as written. The Golomb-ruler claim is stated without a detailed coordinate proof, and the quantitative conclusion that applying b_k instead of b to Theorem 5 gives a contradiction at n = 493 and c = 12 is asserted without showing the inequality or the arithmetic. Algorithm 3, the only computational evidence for the value 493, has serious presentation gaps: the variable `ints` in the initialization of `sum` is undefined, the loop sets blue_array[3] = 3 although the largest Golomb-ruler order fitting in length 2 is 2, and the greedy loop is not shown to compute a valid lower bound for the left-hand side of the Theorem 5 inequality when the allowed number of intervals depends on the subdiagonal index k. A complete proof or a shipped, runnable script with exact integer arithmetic is needed.
  3. [§2.3, Theorem 5] Theorem 5's claim that n = 772 and c = 12 yield a contradiction is supported only by the sentence that the two displayed conditions cannot both be true. Unlike Theorems 3 and 4, which at least refer to Python scripts, no computation or script is supplied for this load-bearing check. The paper should include the arithmetic or a reproducible program, or give a hand-checkable derivation of the 772 bound.
minor comments (5)
  1. [Abstract and §2.1] The abstract says the bound is improved first to 1803, but Theorem 3 and the introduction state R_3(L) <= 1804; the discrepancy should be reconciled.
  2. [§2.3, Algorithm 2] The line `smin = intvls + 2 + 2(b-1)*C(q,2) + qr` would be much easier to check if the comment noted that it equals n* + floor((n*-1)/2) + 1 plus the same optimization terms from the proof of Theorem 4.
  3. [§3, Theorem 8] The proof refers to 'Lemma 11 (number may change)', which is a leftover editorial note and should be replaced by the correct lemma number.
  4. [§3.1.4] The sentence 'fixing these values in this took much longer' should be rewritten, and the conflict counts from different solver runs should be presented with the solver version and run configuration for comparability.
  5. [§2.1] The sentence 'we have ceil(n/3) of them and will prove this later' is not a proof; the pigeonhole argument from Theorem 2 should be restated at the point where it is first used.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the upper-bound chain uses external theorems, standard Golomb-ruler data, and numerical computations that do not encode the claimed bounds.

full rationale

The paper does not derive R_3(L) <= 493 from a prior assumption of that bound. Theorem 2 is taken from Canacki et al. as an external starting point (R_3(L) <= 2593). Theorems 3 and 4 set up independent counting inequalities (s_min > n and s*_min > 2n) and report numerical thresholds 1804 and 1573; those thresholds are outputs of the stated inequalities, not inputs. Theorem 5 asserts that for n=772 and c=12 the two displayed inequalities cannot both hold; while the arithmetic is not shown in detail, that is a computational claim rather than a circular one. Theorem 6 replaces the uniform forcing bound b with a Golomb-ruler order bound b_k; the Golomb-ruler data are standard external values, and the argument that equal spacings among forced blue points yield a monochromatic L is a structural forcing argument independent of the target value. The lower-bound and SAT-solver material is separate and does not feed the upper-bound proof. No fitted parameters are renamed as predictions, and no load-bearing step reduces an equation to itself. The acknowledgements disclose that referees supplied the 772 and 493 proofs; that is transparency about external verification, and the abbreviated presentation of Theorems 5-6 and Algorithm 3 is a reproducibility concern rather than circularity. Accordingly the circularity score is 0.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The central claim rests on standard Ramsey-theory facts, a geometric forcing property inherited from prior work, and a table of optimal Golomb rulers. No free parameters are fitted to data; the numerical bounds come from finite optimization computations.

assumptions (3)
  • standard math Gallai-Witt theorem guarantees existence of a monochromatic L for sufficiently large n.
    Invoked as Theorem 1 to frame the problem and establish that a finite R_3(L) exists.
  • domain assumption Geometric forcing property: if a point p is forced by a red pair to be non-red, then a point q completing an L with p and another such forced point cannot be red or the forced color without creating a monochromatic L.
    Used throughout Theorems 2 and Lemma 1; taken from the Canacki et al. proof without a standalone derivation in this paper.
  • domain assumption Known optimal Golomb ruler lengths for orders 0 through 28.
    Used in Theorem 6 and Algorithm 3 to bound the number of forced points on a subdiagonal; no source is cited for these values.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Ramsey Theory on the Integer Grid: The "L" Problem." pith.science (2026). https://pith.science/paper/DCK7K4SC

@misc{pith2026250205162,
  author       = {Pith},
  title        = {Pith review of: Ramsey Theory on the Integer Grid: The "L" Problem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DCK7K4SC}},
  note         = {Machine review of arXiv:2502.05162}
}
abstract

In an $[n] \times [n]$ integer grid, a monochromatic $L$ is any set of points $\{(i, j), (i, j+t), (i+t, j+t)\}$ for some positive integer $t$, where $1 \leq i, j, i+t, j+t \leq n$. In this paper, we investigate the upper bound for the smallest integer $n$ such that a $3$-colored $n \times n$ grid is guaranteed to contain a monochromatic $L$. We use various methods, such as counting intervals on the main diagonal and using Golomb rulers, to improve the upper bound. This bound originally sat at 2593, and we improve it first to 1803, then to 1573, then to 772, and finally to 493. In the latter part of this paper, we discuss the lower bound and our attempts to improve it using SAT solvers.

Figures

Figures reproduced from arXiv: 2502.05162 by the authors.

Figure 1
Figure 1. An illustration of the contradiction we reach in Theorem 1. P [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. An illustration of how certain intervals force points on corr [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. A satisfiable 20 × 20 grid as found by CryptoMiniSAT. If the integer in the ith row and jth column is c, then the color of (i, j) in our grid is color c. In this grid (as well as all subsequent grids in this paper), the 0s are colored red, the 1s are colored blue, and the 2s are colored green. Theorem 7. The number of CNF statements for checking whether a 3−colored n × n grid has no monochromatic L is n(n−1)(2n−1) 2… view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: A representation of an n × n grid, where each point p is labeled with the number of Ls that can be formed with p as the uppermost point. 3.1.1. Brute Force This method briefly involves taking the entire CNF statement that corresponds with a 3-colored 21 × 21 grid with …
Figure 5
Figure 5. Figure 5: A picture of the main reverse diagonal (colored red) on a 5 [PITH_FULL_IMAGE:figures/full_fig_p017_5.png]
Figure 6
Figure 6. Figure 6: A table of the solving methods mentioned in 5 [PITH_FULL_IMAGE:figures/full_fig_p018_6.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

10 extracted references · 10 canonical work pages

  1. [3]

    S. Cai, Z. Chen, and X. Zhang, ParKissat, accessible at https://github.com/shaowei-cai-group/ParKissat-RS

    work page

  2. [1]

    Balaji, A

    V. Balaji, A. Lott, and A. Rice, Schur’s theorem in intege r lattices, Integers 22 (2022), #A62

    work page 2022

  3. [2]

    On SAT Solvers and Ramsey-type Numbers

    B. Canacki, H. Christenson, R. Fleischman, N. McNabb, an d D. Smolyak, On sat solvers and Ramsey-type numbers, preprint, arXiv: 2312.01159

    work page Pith review arXiv

  4. [4]

    Dumitrescu and R

    A. Dumitrescu and R. Radoicic, On a coloring problem for t he integer grid, Contemp. Math. 342 (2004), 67–74

    work page 2004

  5. [5]

    R. L. Graham, B. L. Rothschild, and J. H. Spencer, Ramsey Theory, John Wiley & Sons, New York, 1991

    work page 1991

  6. [6]

    R. L. Graham and J. Solymosi, Monochromatic equilateral right triangles on the integer grid, in Topics in Discrete Mathematics: Dedicated to Jarik Neˇ setˇ ril on the Occasion of his 60th Birthday, Springer Berlin Heidelberg, 2006

    work page 2006

  7. [7]

    Manske, Coloring in Extremal Problems in Combinatorics , Iowa State University, Ames, 2010

    J. Manske, Coloring in Extremal Problems in Combinatorics , Iowa State University, Ames, 2010

    work page 2010

  8. [8]

    I. D. Shkredov and J. Solymosi, Titled corners in integer grids, in Number Theory and Com- binatorics: A Collection in Honor of the Mathematics of Rona ld Graham , de Gruyter, 2022

    work page 2022

Show all 10 references
  1. [9]

    Soos, Cryptominisat, accessible at https://msoos.github.io/cryptominisat_web/

    M. Soos, Cryptominisat, accessible at https://msoos.github.io/cryptominisat_web/

  2. [10]

    Witt, Ein kombinatorischer Satz der Elementargeome trie, Mathematische Nachrichten 6 (1952), 261-262

    E. Witt, Ein kombinatorischer Satz der Elementargeome trie, Mathematische Nachrichten 6 (1952), 261-262

    1952

Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.