{"id":"c975b99a-3bfe-4744-8d88-8750a3f6725f","arxiv_id":"2502.05162","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Any 3-coloring of a 493 by 493 integer grid must contain a monochromatic L-shape, improving the previous upper bound of 2593.","lead":"This paper improves the known upper bound on the size of a grid that must contain a monochromatic L-shaped pattern in any 3-coloring, from 2593 down to 493. The authors use careful counting of gaps between same-colored diagonal points and a connection to Golomb rulers.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The 493 bound is not established by the written proof: Theorem 6 is a one-paragraph assertion and Algorithm 3 has undefined variables, an unexplained initial term, and an off-by-one in the Golomb array, so the decisive computation is not verifiable.","rationale":"The reader correctly located the Golomb-ruler step as the weakest point. My read agrees that this is where the proof is most fragile, but I do not think the Golomb-ruler lemma itself is false: the missing contradiction can be supplied by taking two same-difference blue pairs on a subdiagonal and forming the cross-blue, cross-red, and monochromatic-green configurations. The more serious problem is that the final numerical value 493 is produced by an unshipped, partially undefined algorithm, and the transition from the repaired lemma to 493 is not a step the reader can check from the text. This does not establish that the theorem is false; it establishes that the paper, as presented, has a verification gap in its central claim. That matches the reader's CONDITIONAL verdict, so no adjustment is needed. The proposed concrete check is a direct re-implementation of the optimization with explicit constraints, which would settle whether 493 is actually obtained or whether the published number depends on an artifact of the pseudocode.","tokens_in":14043,"tokens_out":52309,"duration_ms":493980,"concrete_test":"Independently recompute the claimed threshold: for n = 493 and c = 12, minimize sum_{j=1}^{c} sum_{i=1}^{n* - j} a_{i,i+j} + c n* subject to a_{i,i+j} >= j-1 and, for each absolute distance v+1, at most 2 * b_{v+1} red pairs, where b_t is the largest Golomb-ruler order of length n-t-1. Then compare this minimum with n*c(c+1)/2. Repeat for decreasing n to find the least n for which the minimum exceeds the right-hand side; if that least n is greater than 493, the numerical claim of Theorem 6 is unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim R_3(L) ≤ 493 rests on the statement in Theorem 6 that replacing the uniform bound b by the position-dependent Golomb-order bound b_k in Theorem 5 'gives us a contradiction when n = 493 and c = 12'. The qualitative Golomb-ruler idea is plausible and can be repaired: if two blue pairs on the same S_k have the same difference, the cross configuration forces a point that cannot be colored red, blue, or green. But the quantitative step from that idea to 493 is not demonstrated in the paper. Algorithm 3, which is the only evidence for the number 493, contains serious presentation gaps: the variable `ints` is undefined; the initial value assigned to `sum` is not derived from the inequality in Theorem 5; and the construction of `blue array` has an off-by-one. For example, at index i=3 the loop yields blue_array[3]=3, whereas the largest Golomb-ruler order that fits in length 2 is order 2. The paper gives no proof that the greedy loop in Algorithm 3 computes a valid lower bound for the left-hand side of the Theorem 5 inequality when the count bound varies with the subdiagonal. Since the theorem's only route to n=493 is this computation, the claim is not independently checkable from the manuscript as written.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":14286,"tokens_out":14839,"duration_ms":128070,"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":[{"comment":"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.","section":"§2.1, Lemma 1"},{"comment":"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.","section":"§2.3, Theorem 6 and Algorithm 3"},{"comment":"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.","section":"§2.3, Theorem 5"}],"minor_comments":[{"comment":"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.","section":"Abstract and §2.1"},{"comment":"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.","section":"§2.3, Algorithm 2"},{"comment":"The proof refers to 'Lemma 11 (number may change)', which is a leftover editorial note and should be replaced by the correct lemma number.","section":"§3, Theorem 8"},{"comment":"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.","section":"§3.1.4"},{"comment":"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.","section":"§2.1"}],"recommendation":"major_revision","confidential_remarks":"The paper is a good fit for a combinatorics journal and the qualitative ideas are promising, but the lack of reproducible computation is the main barrier. If the authors correct the b formula and supply complete proofs or verified scripts for the 772 and 493 bounds, I would expect the central claims to be checkable and the paper to become publishable. The internal inconsistencies, such as the 1803/1804 mismatch and the b = 61 statement versus the printed formula, suggest the numerical results were not fully checked against the displayed equations."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe paper is worth a look if you care about grid Ramsey numbers, but be warned: the headline bound, R_3(L) ≤ 493, is not actually established by the written proof. The route from 2593 down to 1804 and 1573 is new and plausible—the interval-counting framework is a genuine idea—but the last two steps, to 772 and 493, are asserted as computations without enough detail to check. Theorem 6 in particular is a one-paragraph argument plus a pseudocode that has undefined variables and an off-by-one in the Golomb-ruler array. The qualitative Golomb-ruler idea is attractive and probably repairable, but the 493 number comes from that algorithm, and the paper gives no proof that the greedy loop computes a valid lower bound.\n\nCredit where it's due: the paper is clearly written, the interval method is a real improvement, and the lower-bound/SAT section is honest about what didn't work. The early theorems' formulas are explicit enough that a patient reader could verify 1804 and 1573 with their own script. So this isn't junk.\n\nWhat's missing: code or tables for the computational claims, a thorough proof of the Golomb-ruler forcing property, and a fix for Algorithm 3. Without those, the two best numbers (772, 493) are conjectures rather than theorems. That's a load-bearing gap, because the entire point is the bound.\n\nBottom line: this paper deserves a serious referee, not a desk reject. A referee could ask for the missing artifacts and a corrected algorithm, and perhaps the result will survive. I wouldn't cite the 493 bound in my own work yet.","headline":"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.","tokens_in":14786,"tokens_out":11486,"would_cite":false,"duration_ms":101564,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05D10"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["Ramsey theory","integer grid","monochromatic L","Golomb ruler","interval counting","SAT solver","3-coloring"],"falsifier":"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.","tokens_in":13830,"feed_emoji":"📐","tokens_out":9895,"duration_ms":79980,"temperature":0.7,"pith_summary":"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.","feed_headline":"Every 3-coloring of a 493x493 grid contains a monochromatic L.","feed_subtitle":"New proof using diagonal counting and Golomb rulers shrinks the upper bound from 2593.","key_machinery":"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.","core_discovery":"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 $n=493$.","pith_inferences":["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."],"forward_implications":["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 $R_3(L) \\ge 14$."],"supporting_citations":[{"why":"Supplies the previous upper bound of 2593 and the forcing framework that the paper's interval-counting improvements build on.","marker":"[2]"},{"why":"Provides the existence theorem guaranteeing that a monochromatic L appears in every k-coloring of a large enough grid, the starting point of the problem.","marker":"[10]"},{"why":"Gives the known value $R_2(L)=5$ used to frame the focus on $R_3(L)$.","marker":"[7]"}],"fun_headline_variants":["Ramsey L bound drops to 493 via Golomb rulers","Any 3-coloring of a 493x493 grid contains a monochromatic L","Golomb rulers cut the L-number from 2593 to 493","New proof: every 3-color 493x493 grid hides an L shape","The L problem: 493 is the new upper bound"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Ramsey L bound drops to 493 via Golomb rulers","Any 3-coloring of a 493x493 grid contains a monochromatic L","Golomb rulers cut the L-number from 2593 to 493","New proof: every 3-color 493x493 grid hides an L shape","The L problem: 493 is the new upper bound"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000235,"raw_usage":{"total_tokens":1461,"prompt_tokens":870,"completion_tokens":591,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":486,"completion_tokens_details":{"reasoning_tokens":494}},"tokens_in":486,"tokens_out":591,"duration_ms":5553,"temperature":1.0,"reasoning_tokens":494,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T20:03:52.290287+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"On SAT Solvers and Ramsey-type Numbers","cited_arxiv_id":"2312.01159","evidence_quote":"Supplies the previous upper bound of 2593 and the forcing framework that the paper's interval-counting improvements build on."},{"cited_title":"Witt, Ein kombinatorischer Satz der Elementargeome trie, Mathematische Nachrichten 6 (1952), 261-262","cited_arxiv_id":null,"evidence_quote":"Provides the existence theorem guaranteeing that a monochromatic L appears in every k-coloring of a large enough grid, the starting point of the problem."},{"cited_title":"Manske, Coloring in Extremal Problems in Combinatorics , Iowa State University, Ames, 2010","cited_arxiv_id":null,"evidence_quote":"Gives the known value $R_2(L)=5$ used to frame the focus on $R_3(L)$."}],"review_version":1}