{"id":"c44c265d-555f-463e-8dad-add83f640021","arxiv_id":"2507.05231","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"low","formal_verification":"none","parameter_count":1,"one_line_summary":"A new construction lowers the triangle-removal-lemma lower-bound exponent constant from about 0.83 to about 1.66, using a Euclidean-ball sumset estimate.","lead":"This paper constructs graphs that require many edge deletions to become triangle-free but contain very few triangles, improving the best known constant in the triangle removal lemma by a factor of two. The construction borrows a geometric trick from additive combinatorics: using a high-dimensional Euclidean ball instead of a box to make sums stay inside with higher probability.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 2's stated constant C ≈ 1.6601 is arithmetically wrong: 1/(4 log_2(4/3)) ≈ 0.602, not 1.660; the proof yields the smaller, correct constant.","rationale":"The paper's mathematical argument is essentially sound: the reduction from η(n) to δ(ε) is standard, the discretization via Lemmas 3.2–3.3 works, and Proposition 3.1—the ball probability estimate—is proved correctly modulo a sign typo in Lemma 3.5 (the displayed inequality should have +2λ_xλ_y⟨x̂,ŷ⟩, and since ⟨x̂,ŷ⟩ ≤ −1/2 the conclusion still follows). The optimized constants from Theorem 1 are correct: the (3/4)^{D/2} range gives B = 2√(log_2(4/3)), and therefore C = 1/(4 log_2(4/3)) ≈ 0.602. The previous best from Green's corners construction had B_old = 2√(2 log_2(4/3)), so C_old = 1/(8 log_2(4/3)) ≈ 0.301, exactly half of the new C. The problem is that the paper's decimal approximations are wrong: it prints C ≈ 1.6601, which is actually the denominator 4 log_2(4/3), and it prints the old constant as ≈ 0.8301, which is actually 2 log_2(4/3). These errors appear in the abstract, the introduction, and Theorem 2, and they misstate the strength of the main result by a factor of about 2.76. This is a load-bearing issue because the quantitative claim is the paper's central contribution; however, it is a fixable numerical error, not a flaw in the proof. The reader's verdict of CONDITIONAL is therefore appropriate, and our independent concern reinforces the need for correction rather than moving the verdict to accept or reject.","tokens_in":10929,"tokens_out":32642,"duration_ms":297509,"concrete_test":"Recompute the constant: log_2(4/3) = ln(4/3)/ln 2 ≈ 0.41504, so 4 log_2(4/3) ≈ 1.66016 and C = 1/(4 log_2(4/3)) ≈ 0.60236. Check that the abstract's '≈1.6601' equals the denominator, not C. Independently re-derive C from Theorem 1 by optimizing D in (3/4)^{D/2} n^{-2/D}: the optimal D is 2/√(log_2(4/3)), giving coefficient 2√(log_2(4/3)) in the η exponent, hence C = 1/(4 log_2(4/3)). If the printed approximations are not corrected, the theorem statement as numerically stated is false.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim's numerical constants are internally inconsistent. In Theorem 2 and the abstract, the paper states C = 1/(4 log_2(4/3)) ≈ 1.6601, and the previous best as 1/(2 log_2(4/3)) ≈ 0.8301. But log_2(4/3) = ln(4/3)/ln 2 ≈ 0.4150, so 4 log_2(4/3) ≈ 1.660 and 1/(4 log_2(4/3)) ≈ 0.602, not 1.660. The number 1.660 is the denominator, not C. Similarly, 1/(2 log_2(4/3)) ≈ 1.204, while 2 log_2(4/3) ≈ 0.830; the printed 0.830 is actually 2 log_2(4/3). The proof itself (Theorem 1 with the new (3/4)^{D/2} range) yields, after optimizing D, an exponent coefficient B = 2√(log_2(4/3)) in η(n) ≳ 2^{-B√(log_2 n)}, and hence δ(ε) ≤ ε^{C log_2(1/ε)} with C = 1/B^2 = 1/(4 log_2(4/3)) ≈ 0.602. The previous Green range (3/4)^D gives B_old = 2√(2 log_2(4/3)) and C_old = 1/(8 log_2(4/3)) ≈ 0.301. Thus the correct improvement is a factor of 2 (0.301 → 0.602), but the numbers printed as 1.660 and 0.830 are each wrong by a factor of about 2.76. A reader relying on the approximations would believe in a far stronger bound than proven. The symbolic formulas are correct; the decimal evaluations are not. This is directly about the central claim, and the sign typo in Lemma 3.5 and the missing factor in Corollary 2.4 are additional but less severe defects.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs n-vertex graphs with few triangles but many edges that must be deleted in a triangle-free subgraph, thereby giving lower bounds for the triangle removal lemma. It proves a new 'range' of bounds η(3n) ≫ D^{-O(1)}(3/4)^{D/2} n^{-2/D} for arbitrary D, improving the previous Green-type range (3/4)^D. Optimizing D gives η(n) ≫ 2^{-(2√(log_2(4/3))+o(1))√log_2 n} and hence δ(ε) ≤ ε^{(C-o(1)) log_2(1/ε)} with C = 1/(4 log_2(4/3)). The source of the improvement is Proposition 3.1, which replaces the box of side roughly M with a Euclidean ball: the probability that two independent random points of the ball have their sum in the ball is ≫ D^{-O(1)}(3/4)^{D/2}, instead of (3/4)^D for a box. The proof chain uses a reduction from corner-free sets (Lemma 2.1 and Claim 2.2), a Behrend-style coloring (Lemma 2.3), a discretization lemma (Lemma 3.2), and an explicit integration bound. The paper also proves a limitation result (Proposition 4.1) showing that the more general Proposition 1.2 is polynomially related to the 3-AP-free density r3(n)/n.","tokens_in":11415,"tokens_out":27548,"duration_ms":249436,"significance":"After correcting the arithmetic errors noted below, the paper improves the best-known exponent coefficient in the lower bound for 1/δ(ε) from 1/(8 log_2(4/3)) ≈ 0.301 to 1/(4 log_2(4/3)) ≈ 0.602, a factor of 2. This is a modest but genuine advance over the previous corner-based bound of Green. The proof is elementary, self-contained, and has no fitted parameters; the central geometric estimate is explicit. The limitation discussion in Section 4 is honest and connects the construction to known bounds for r3(n). The self-citation [5] is not used in the proof, so there is no circularity. The paper is a short note that fits well in a combinatorics venue, provided the numerical and proof-presentation issues are fixed.","major_comments":[{"comment":"The decimal evaluations of the constants are arithmetically wrong. C = 1/(4 log_2(4/3)) is approximately 0.602, not 1.6601; the value 1.6601 is instead 4 log_2(4/3). Similarly, 1/(2 log_2(4/3)) is approximately 1.204, not 0.8301, and the previous exponent coefficient from Eq. (1.2) is actually C_old = 1/(8 log_2(4/3)) ≈ 0.301, with C_new = 2·C_old. As printed, the abstract and Theorem 2 overstate the improvement by a factor of about 2.76. The symbolic formulas are correct, and the proof actually gives the smaller constant, but the headline numerical claims must be corrected.","section":"Abstract and Theorem 2"},{"comment":"The expansion of ||x+y||² is printed as ||x||² − 2⟨x,y⟩ + ||y||²; the correct identity has +2⟨x,y⟩. With the printed minus sign, the inequality λ_x² − 2λ_xλ_y⟨x̂,ŷ⟩ + λ_y² ≤ λ_x² does not follow from ⟨x̂,ŷ⟩ ≤ −1/2. Replacing the minus sign with a plus sign makes the argument work: one obtains λ_x² + λ_y² + 2λ_xλ_y⟨x̂,ŷ⟩ ≤ λ_x² + λ_y² − λ_xλ_y ≤ λ_x² when λ_x ≥ λ_y. Since Lemma 3.5 is the bridge from the ball probability to the inner-product computation (Eqs. 3.1–3.4), the proof as written is invalid, though the fix is local.","section":"Section 3.1, proof of Lemma 3.5"},{"comment":"The lattice-shift step has a gap. The set S is defined as {(x,y): x,y,x+y ∈ B}, and Lemma 3.2 gives lattice points (ξ,η) ∈ (t+S)∩Z^{2D}. For such a point, ξ ∈ t^(1)+B, η ∈ t^(2)+B, and ξ+η ∈ t^(1)+t^(2)+B. The proof then sets Z0 := B∩Z^D and lower-bounds the number of pairs (x,y) ∈ X0×Y0 with x+y ∈ Z0, but these pairs are not directly counted by the measure μ(S). The correct choice is Z0 := (t^(1)+t^(2)+B)∩Z^D, or one must argue that t^(1)+t^(2) is negligible compared with the radius of B (which is plausible under the hypothesis n > (2D)^{100D} and the later bound r ≥ D^{50}). As written, the displayed lower bound '#(x∈X0,y∈Y0: x+y∈Z0) ≫ D^{-O(1)}(3/4)^{D/2} n²' is not justified.","section":"Proof of Theorem 1 (Section 3)"}],"minor_comments":[{"comment":"Lemma 2.1 states η(3n) ≥ |A|/(9n²), but the constructed graph has 3|A| edges because each triangle contributes three edges and the triangles are edge-disjoint; hence one actually gets η(3n) ≥ |A|/(3n²). This constant factor does not affect the asymptotic exponent, but it propagates to Proposition 1.2 and Corollary 2.4, which are accordingly weaker by a factor of 3.","section":"Lemma 2.1 and Corollary 2.4"},{"comment":"The displayed formula for D_n is hard to parse and appears to read 2√(log_2(4/3))√(log_2 n). If that is the intended choice, substituting it into (3/4)^{D/2} n^{-2/D} gives an exponent of about (a^{3/2}+a^{-1/2})√log n with a = log_2(4/3), i.e., about 1.82√log n, not the claimed 2√(log_2(4/3))√log n. The correct optimizing choice is D_n ≈ 2√(log_2 n / log_2(4/3)). Please clarify or correct the formula.","section":"Proof of Theorem 1, optimizing D"},{"comment":"There are several typographical errors: 'Vinagradov' should be 'Vinogradov'; the reference to 'Proposition 2.4' after Lemma 3.2 should be to Corollary 2.4; and the sentence 'It is clear that Eq. 3.2 holds' appears to refer to the equality to P(⟨u,v⟩ ≤ −1/2), not to the inequality Eq. 3.1.","section":"Throughout"},{"comment":"The phrase 'the previous best bound had a worse constant of 1/(2 log_2(4/3)) ≈ 0.8301' is inconsistent both arithmetically and with the abstract's definition C_old := C_new/2. The previous bound from Eq. (1.2) has C_old = 1/(8 log_2(4/3)) ≈ 0.301; please correct the displayed expression and its decimal evaluation.","section":"Section 1, introduction"}],"recommendation":"major_revision","confidential_remarks":"The manuscript contains several arithmetic and typographical errors in its headline constants, but the underlying mathematics is sound and the fixes are local. I recommend major revision rather than rejection: the central derivation (reduction, coloring, ball discretization, optimization) is correct after the sign and shift issues are repaired, and the improved constant is a legitimate, if modest, advance. The paper is suitable for a combinatorics journal."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The core idea is clean and correct: replace the D-dimensional box with a Euclidean ball to change the closure probability from (3/4)^D to (3/4)^(D/2), and the optimization then genuinely improves the triangle removal bound by a factor of two. Theorem 1 and the derivation of Theorem 2 are sound; Proposition 1.2 is a nice abstraction, and the discussion in Section 4 is thoughtful. I checked the main calculation and the symbolic statement C = 1/(4 log_2(4/3)) is right, and the previous best C_old = 1/(8 log_2(4/3)) is also right. The constant is approximately 0.602, not 1.660; the old one is approximately 0.301, not 0.830. The abstract and Theorem 2 printed the denominators as the constants. That is not a minor typo: a reader relying on the approximations will think the bound is far stronger than proven. It should be fixed before publication, and the author should re-check all numeric evaluations. The other issues are smaller. Lemma 3.5 has a sign typo in ||x+y||^2 (should be +2< x,y >, not -2< x,y >); the argument still works because < x,y > is negative. Corollary 2.4 drops a factor 9 from Lemma 2.1, but it is absorbed by implicit constants later. Neither affects the main theorem. The citation pattern looks honest: the author's own work [5] is mentioned but not used, and the referenced claims (Ruzsa, BSG, Tao-Vu) are standard. There is no fitting or circularity. This paper deserves a serious referee. It is a short, significant contribution to a central problem, with a new geometric idea that is likely to be quoted. I would recommend acceptance after the numerical errors are corrected. I would bring it to reading group, and I would cite it once the constants are fixed.","headline":"The ball-for-box swap gives a real factor-two improvement and the proof is sound, but the abstract and Theorem 2 misstate the numerical constants (C≈0.602, not 1.660).","tokens_in":628,"tokens_out":657,"would_cite":true,"duration_ms":44468,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C35","11B25","05C65"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper constructs graphs that double the best-known lower-bound exponent for the triangle removal lemma, reaching C\\approx 1.6601 instead of C\\approx 0.8301.","keywords":["triangle removal lemma","corner-free sets","progression-free sets","Euclidean ball","additive combinatorics","random unit vectors","diamond-free graphs","graph removal bounds"],"falsifier":"Evaluate the closure probability of the centered Euclidean ball numerically for large dimension, say $D=100$, by Monte Carlo sampling or by direct integration of $\\int_{-1/2}^{-1}(1-r^2)^{D/2}\\,dr$. If the observed decay follows $(3/4)^D$ rather than $(3/4)^{D/2}$, then Proposition 3.1 is false and the improved constant in Theorem 2 cannot stand; a rigorous asymptotic of that integral that contradicts $\\gg D^{-O(1)}(3/4)^{D/2}$ would settle it the same way.","tokens_in":10712,"feed_emoji":"📐","tokens_out":17157,"duration_ms":166657,"temperature":0.7,"pith_summary":"The paper aims to improve the best-known lower bound for the triangle removal lemma, which says that a graph with fewer than $\\delta n^3$ triangles can be made triangle-free by deleting fewer than $\\varepsilon n^2$ edges. The paper constructs graphs that are far from triangle-free yet contain very few triangles: $\\varepsilon^{(C-o(1))\\log_2(1/\\varepsilon)}n^3$, with $C\\approx 1.6601$, doubling the previously known constant $C\\approx 0.8301$. The reason this works is geometric: earlier constructions used a $D$-dimensional box, whose two-random-points-sum-to-a-third probability is $(3/4)^D$, while the new construction uses a Euclidean ball, whose probability is $(3/4)^{D/2}$. Optimizing the dimension converts that square-root saving into the improved exponent, narrowing the gap between the tower-type upper bound and the $\\exp(\\log^2)$ lower bound for the inverse removal function.","feed_headline":"Ball geometry doubles the triangle removal bound","feed_subtitle":"The Euclidean ball improves the best-known constant from about 0.83 to about 1.66.","key_machinery":"The load-bearing object is a corner-free set inside a product grid: a set with no triple $(x,y),(x+d,y),(x,y+d)$ for $d\\neq 0$. Lemma 2.1 converts such a set into a tripartite graph whose triangles are edge-disjoint, so the graph is an instance of the $\\eta(n)$ extremal problem; Corollary 2.4 reduces the density to the additive concentration probability $\\Pr_{x\\sim X,y\\sim Y}(x+y\\in Z)$. The paper's new choice for $X,Y,Z$ is a discretization of the Euclidean ball, and Proposition 3.1 supplies the estimate $\\Pr_{x,y\\sim B}(x+y\\in B)\\gg D^{-O(1)}(3/4)^{D/2}$ via the known density of the dot product of two random unit vectors. This estimate, rather than the box's $(3/4)^D$, is what carries the improved exponent after optimizing $D$.","core_discovery":"On the paper's own terms, the discovery is Theorem 2: $\\delta(\\varepsilon) \\le \\varepsilon^{(C-o(1))\\log_2(1/\\varepsilon)}$ for $C = 1/(4\\log_2(4/3)) \\approx 1.6601$, upgrading the previously known constant $C_{\\text{old}} \\approx 0.8301$. Equivalently, Theorem 1 gives $\\eta(3n) \\gg D^{-O(1)}(3/4)^{D/2} n^{-2/D}$ for every dimension $D$, which optimizes to $\\eta(n) \\gg 2^{-(2\\sqrt{\\log_2(4/3)}+o(1))\\sqrt{\\log_2 n}}$. The engine is the Euclidean ball: a $D$-dimensional box gives only a $(3/4)^D$ closure probability, while the ball gives $(3/4)^{D/2}$ (Proposition 3.1), and that square-root saving is exactly what doubles the exponent in the final bound. The paper also records a general reduction (Proposition 1.2) from additive sets with a 3-AP-free difference set to such graphs, and a converse estimate (Proposition 4.1) showing that this particular reduction cannot by itself produce dramatically stronger bounds.","pith_inferences":["The same 'ball instead of box' move could plausibly improve lower bounds for other removal lemmas whose constructions are built from additive concentration, such as the $k$-clique removal lemma, by replacing the box closure probability with the ball's.","A sharper asymptotic for the closure probability $\\Pr_{x,y\\sim B}(x+y\\in B)$ than the lower bound in Proposition 3.1 might move the constant further; the paper's $D^{-O(1)}$ factor leaves room.","The paper's converse estimate suggests that to go beyond this exponent through additive-set reductions one would need new bounds for 3-AP-free sets themselves; the geometric improvement is already tight within that reduction.","One could test the mechanism numerically by estimating the decay exponent of the closure probability for $\\ell^p$ balls with $p\\neq 2$, to see whether the Euclidean ball is optimally closed under addition among convex bodies."],"forward_implications":["If Theorem 2 is correct, the best-known lower bound on the inverse triangle removal function becomes $1/\\delta(\\varepsilon) \\gg \\exp((C-o(1))\\log(1/\\varepsilon)\\log_2(1/\\varepsilon))$ with $C\\approx 1.6601$, the strongest known.","The construction supplies $n$-vertex graphs that are $\\varepsilon$-far from triangle-free while containing only $\\varepsilon^{(C-o(1))\\log_2(1/\\varepsilon)} n^3$ triangles, for the same constant.","The dimension-range formulation $\\eta(3n)\\gg D^{-O(1)}(3/4)^{D/2}n^{-2/D}$ means the improvement is robust to the choice of dimension and is optimized near $D \\sim 2\\sqrt{\\log_2 n/\\log_2(4/3)}$.","For every construction coming from this additive-combinatorial template, the constant is controlled by the closure probability of the chosen geometric body, so the improved exponent is a direct corollary of the ball's square-root saving over the box."],"supporting_citations":[{"why":"Supplies the classical grid construction of progression-free sets, the template for the dimension-range bounds, and the lower bound $r_3(n)\\ge n^{1-o(1)}$ used in the limitations section.","marker":"[1]"},{"why":"Establishes the corners-problem range $(3/4)^D$ that was the previous best; Theorem 1 improves exactly this range to $(3/4)^{D/2}$.","marker":"[9]"},{"why":"Earlier refinement of the corners problem that introduced the range-of-bounds viewpoint and motivates the proof structure.","marker":"[14]"},{"why":"Gives the exact density of the dot product of two random unit vectors, which the proof of Proposition 3.1 uses to estimate the ball's closure probability.","marker":"[15]"}],"fun_headline_variants":["Ball geometry doubles triangle removal constant","Triangle removal lemma sharpened via Euclidean ball","How a ball beats a box in triangle removal"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole improvement rests on the estimate that two independent random points in a high-dimensional Euclidean ball have their sum inside the ball with probability at least $D^{-O(1)}(3/4)^{D/2}$; if the true probability decayed as $(3/4)^D$, optimizing the dimension would return the old, unimproved constant.","fun_headline_variants_meta":{"raw":{"variants":["Ball geometry doubles triangle removal constant","Triangle removal lemma sharpened via Euclidean ball","How a ball beats a box in triangle removal"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000275,"raw_usage":{"total_tokens":1635,"prompt_tokens":927,"completion_tokens":708,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":543,"completion_tokens_details":{"reasoning_tokens":665}},"tokens_in":543,"tokens_out":708,"duration_ms":8543,"temperature":1.0,"reasoning_tokens":665,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T19:31:09.517817+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate the closure probability of the centered Euclidean ball numerically for large dimension, say $D=100$, by Monte Carlo sampling or by direct integration of $\\int_{-1/2}^{-1}(1-r^2)^{D/2}\\,dr$. If the observed decay follows $(3/4)^D$ rather than $(3/4)^{D/2}$, then Proposition 3.1 is false and the improved constant in Theorem 2 cannot stand; a rigorous asymptotic of that integral that contradicts $\\gg D^{-O(1)}(3/4)^{D/2}$ would settle it the same way.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the classical grid construction of progression-free sets, the template for the dimension-range bounds, and the lower bound $r_3(n)\\ge n^{1-o(1)}$ used in the limitations section."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the corners-problem range $(3/4)^D$ that was the previous best; Theorem 1 improves exactly this range to $(3/4)^{D/2}$."},{"cited_title":"Linial and A","cited_arxiv_id":null,"evidence_quote":"Earlier refinement of the corners problem that introduced the range-of-bounds viewpoint and motivates the proof structure."},{"cited_title":"Scalar product of random unit vectors","cited_arxiv_id":null,"evidence_quote":"Gives the exact density of the dot product of two random unit vectors, which the proof of Proposition 3.1 uses to estimate the ball's closure probability."}],"review_version":1}