{"id":"e50e4647-2f69-4311-9ef8-e8d0e772337c","arxiv_id":"1909.00514","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Every graph on n vertices with minimum degree at least ((7+sqrt(21))/14)n admits a fractional triangle decomposition, improving prior bounds.","lead":"This paper lowers the minimum degree needed for a fractional triangle decomposition of a dense graph from 90% to about 82.7% of the vertices. It is a significant step toward Nash-Williams' 1970 conjecture that 75% should suffice.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof of Theorem 1.4 depends on the optimization chain P1→P10, and the printed Lemma 5.15.2 and Lemma 5.17 contain sign errors that break the chain as written; the intended inequalities appear true and repairable, but the gap is real.","rationale":"I read the main theorem as resting on the reduction of the fractional triangle decomposition problem to the optimization program (P1), followed by the chain of upper bounds through P10. The graph-theoretic setup in Sections 3 and 4 is coherent: the delegation and cancellation formalism gives a well-defined weighting, and the transition to the program appears faithful. The concentrated risk is in Section 5, where variable reductions are justified by monotonicity claims. The two flagged statements are indeed wrong as printed: Claim 5.15.2 has a sign error in the derivative of H(s,t), and Lemma 5.17 both reverses a monotonicity inference and asserts a false inequality chain involving 1−5d. Since these lemmas are used to conclude OPT(P1) ≤ OPT(P10) = 3d(1−d)/(1−2d)^2 ≤ 1, the printed proof has a genuine gap at exactly the point where the numerical constant is obtained. However, I checked the intended inequalities: with d = (7−√21)/14, H(a,b) does satisfy H(a,b) ≤ H(0,b), and W10(b) is maximized at b = 0. The errors are localized sign errors in proof presentation rather than evidence that the bound is false. I found no independent fatal flaw in the reduction steps. This matches the reader's weakest-assumption identification, so I recommend no change to the CONDITIONAL verdict: the main claim is plausible and independently supported by the structure of the argument, but the manuscript must be corrected before the theorem is fully established as written.","tokens_in":23108,"tokens_out":13665,"duration_ms":329607,"concrete_test":"Recompute the two disputed inequalities symbolically. For Claim 5.15.2, take H(s,t) = (1−d−s)^2(1−2d−s)/(1−2d−s−t), compute ∂H/∂s by the quotient rule, verify its sign on [0,d]^2 for d = (7−√21)/14, and confirm H(a,b) ≤ H(0,b) by direct evaluation on a fine grid. For Lemma 5.17, recompute F(b) and E(b), check that E is decreasing on [0,d] and that E(b) ≤ E(0) ≤ 0 using the correct value of E(0). If both hold, the chain P1→P10 is repairable with corrected signs; if either fails, Theorem 1.4 is unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is Theorem 1.4, which is reduced to proving OPT(P1) ≤ 1 via the chain of reductions through P10 in Section 5. That chain is the load-bearing part of the argument, and two printed steps are incorrect. In Claim 5.15.2, H(s,t) = (1−d−s)^2(1−2d−s)/(1−2d−s−t); the displayed derivative has the wrong sign. The quotient rule gives ∂H/∂s = H·(−2/(1−d−s) − 1/(1−2d−s) + 1/(1−2d−s−t)) = H·(−2/(1−d−s) + t/((1−2d−s)(1−2d−s−t))), which is negative on the relevant domain. The printed expression is positive, which would imply H(a,b) ≥ H(0,b), the reverse of what Claim 5.15.2 needs. In Lemma 5.17, the text says E(b) is decreasing and therefore E(b) ≥ E(0); the correct inference is E(b) ≤ E(0). It also asserts E(0) ≤ 1−5d ≤ 0 for d ∈ [0,1/5], but 1−5d ≥ 0 there. The correct value E(0) = −1+5d−13d²−12d³ is negative, so the desired conclusion still follows. These are not cosmetic typos in constants: the sign errors invalidate two monotonicity steps that the reduction to P10 relies on. Direct numerical evaluation for d = (7−√21)/14 indicates the intended inequalities are true, so the gap appears localized and repairable rather than fatal, but the manuscript as written does not rigorously establish the chain.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper addresses the fractional version of Nash-Williams' triangle decomposition conjecture. It proves (Theorem 1.4) that every graph on n vertices with minimum degree at least ((7+sqrt(21))/14)n ~ 0.82733n has a fractional K3-decomposition. The proof constructs an explicit triangle weighting w_G by delegating unit demand from each edge through triangles, K4s, and K5s using edge-gadgets; non-negativity of w_G is reformulated as the inequality w_{G,1}(O) <= 1 and then as a sequence of optimization problems (P1) through (P10). Solving (P10) yields the bound 3d(1-d)/(1-2d)^2 <= 1 for d=(7-sqrt(21))/14, which is exactly the stated minimum degree. The paper then uses known iterative-absorption and packing results to derive corollaries for integral triangle decompositions and for packing 2-regular graphs.","tokens_in":23490,"tokens_out":9048,"duration_ms":78579,"significance":"The claimed threshold would be a substantial improvement over the previous best bound delta*_{K3} <= 0.9 (Dross) and a significant step toward the conjectured 0.75 threshold. The construction and main reduction are explicit and hand-checkable, without computer-assisted case analysis (unlike the independent work of Dukes-Horsley), and the delegation/cancellation ideas appear genuinely novel. The paper also gives corollaries for integral decompositions and approximate packings. However, the final monotonicity reductions in Section 5 contain two sign errors that break the chain of inequalities as printed; because these steps are load-bearing for Corollary 5.18 and Theorem 3.12, the manuscript is not yet rigorous in its present form. The errors look localized and numerically repairable, but they must be corrected before the main theorem is established.","major_comments":[{"comment":"The displayed derivative of H(s,t) has the wrong sign on the 2/(1-d-s) term. The logarithmic derivative is dH/ds = H(s,t)(-2/(1-d-s) - 1/(1-2d-s) + 1/(1-2d-s-t)) = H(s,t)(-2/(1-d-s) + t/((1-2d-s)(1-2d-s-t))), which is negative on the relevant domain for the value of d in question. Thus H is decreasing in s, giving H(a,b) <= H(0,b), exactly the inequality Claim 5.15.2 needs. As printed, the calculation uses +2/(1-d-s) and concludes dH/ds >= 0, which would imply the reverse inequality. The claim appears true, but the proof is invalid as written.","section":"Section 5.4, Claim 5.15.2"},{"comment":"The proof has two sign errors. From E'(b) <= 0 it concludes E(b) >= E(0); a decreasing function satisfies E(b) <= E(0). It also states E(0) <= 1-5d <= 0 for d in [0,1/5], but 1-5d is positive for the actual d=(7-sqrt(21))/14 ~ 0.17267. Fortunately E(0) = -1+5d-13d^2-12d^3 is negative at this value, so combining the corrected monotonicity direction with a direct bound on E(0) recovers E(b) <= 0. As printed, however, the proof of G(b) <= G(0) is not rigorous, and this step is necessary for Corollary 5.18.","section":"Section 5.5, Lemma 5.17"}],"minor_comments":[{"comment":"The proof states \"Since a >= d\", but the domain has a in [0,d]; the subsequent inequality is the one that follows from a <= d.","section":"Section 5.3, proof of Claim 5.13.2"},{"comment":"The text says \"we may replace (P8) with a new program (P9)\", but the new program introduced immediately afterwards is labelled (P10); this should say \"replace (P9) with (P10)\".","section":"Section 5.4, after Lemma 5.15"},{"comment":"The text says \"whose minimum is at most that of (P3)\"; since the objective is being replaced by an upper bound, the intended statement is about the maximum.","section":"Section 5.1, introduction of (P4)"},{"comment":"The abstract's final sentence omits the epsilon-slack that appears in Corollary 1.5; as written, \"minimum degree at least 0.82733n\" is not the precise statement of the integral decomposition result.","section":"Abstract"}],"recommendation":"major_revision","confidential_remarks":"The paper is a strong, well-written contribution, and the novel delegation/cancellation method is likely to be influential. The two errors in Section 5 are real but appear localized; I am recommending major revision rather than rejection because the intended inequalities are supported by the surrounding argument and can likely be verified directly. The authors should replace the incorrect derivative and the incorrect monotonicity inference, and add a short verification of the needed bounds for the specific value of d."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick read on 1909.00514. The headline is real: Delcourt and Postle lower the fractional triangle decomposition threshold from 0.9 to 0.82733, beating the independent 0.852 from Dukes-Horsley, and they do it with a hand-verifiable argument rather than computer assistance. The delegation/cancellation scheme is a genuinely new mechanism, and the reduction to a 10-variable nonlinear program is the kind of thing that advances the toolbox for these problems, even if the final optimization is tedious. The corollaries for F-decompositions and cycle packings are straightforward consequences and look sound.\n\nThe proof of Theorem 1.4 rests on Theorem 3.12, which is reduced to solving an optimization chain. Most of the chain is careful and checks out: Propositions 3.11 and 4.7 are clean, and the symmetrization steps in Section 4 are a nice way to collapse many variables to ten. The soft spot is Section 5. As written, Claim 5.15.2 miscomputes the derivative of H(s,t); the sign of the 2/(1-d-s) term is reversed, which inverts the monotonicity conclusion the claim needs. Lemma 5.17 has the same flavor: it says E(b) is decreasing and therefore E(b) >= E(0), when the correct inference is <=, and it follows this with 'E(0) <= 1-5d <= 0' — but 1-5d is nonnegative on the stated interval. The intended inequalities appear to be true (the corrected derivative is negative, and the correct E(0) value is negative for d = (7-√21)/14), so these are repairable sign slips, not a structural hole. But they are load-bearing: they are the steps that drop the program from P9 to P10 and evaluate the final bound. A referee should ask for these to be redone carefully.\n\nOne smaller issue: the abstract overclaims by dropping the epsilon in the integral corollary. The body (Corollary 1.5) correctly states δ >= (7+√21)/14 + ε for triangle decompositions; the abstract says 0.82733n without epsilon. That should be fixed.\n\nOverall: the main theorem is likely correct and the technique is worth knowing. The paper deserves a serious refereeing, with the Section 5 calculations checked line by line. If the authors supply a corrected version of those two lemmas, I would be comfortable taking the result as established.","headline":"Strong new bound on fractional triangle decompositions with a novel delegation/cancellation method; proof has two localized sign slips in the optimization chain that are repairable but must be fixed.","tokens_in":24065,"tokens_out":2293,"would_cite":true,"duration_ms":21459,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C70","05C35","90C30"],"pacs":[],"model":"deepseek-v4-flash","headline":"Every graph with minimum degree above 0.82733n has a fractional triangle decomposition.","keywords":["fractional triangle decomposition","triangle decomposition","minimum degree threshold","edge-gadgets","delegation","cancellation","nonlinear optimization","3-chromatic graphs"],"falsifier":"Evaluate the published objective $\\widehat{W}_9(a,b)$ on a fine grid over $0\\le a,b\\le d$ for $d=(7-\\sqrt{21})/14$. If any value exceeds $1$, the claimed chain of inequalities $\\mathrm{OPT}(P1)\\le \\mathrm{OPT}(P10)=1$ is false, so the proof as printed cannot establish Theorem 1.4.","tokens_in":22868,"feed_emoji":"🔺","tokens_out":13772,"duration_ms":112909,"temperature":0.7,"pith_summary":"The paper tackles the 1970 conjecture that every sufficiently large triangle-divisible graph with minimum degree at least $3n/4$ can have its edges split into edge-disjoint triangles. Because recent work shows the fractional version controls the integral version, the paper proves that every $n$-vertex graph with minimum degree at least $((7+\\sqrt{21})/14)n \\approx 0.82733n$ admits a fractional triangle decomposition. This improves the previous hand-verifiable bound of $0.9n$ and, through a known reduction, gives genuine triangle decompositions for triangle-divisible graphs above this threshold plus any positive epsilon. A sympathetic reader should care because the gap to the conjectured $3/4$ constant is now much smaller, and the same constant bounds decomposition thresholds for all 3-chromatic graphs.","feed_headline":"Fractional triangle-decomposition threshold drops to 0.82733n","feed_subtitle":"The proof's delegation-cancellation scheme narrows the gap to the 75-percent triangle-decomposition conjecture.","key_machinery":"The argument is carried by a weight-distribution scheme built from edge-gadgets. An edge-gadget is a local assignment of weights to the triangles inside a $K_5$ that changes the total weight of exactly one chosen edge while leaving all other edges unchanged. The paper delegates each edge's demand of $1$ uniformly to the triangles containing it, then through the $K_4$'s containing each triangle, then through the $K_5$'s containing each $K_4$; the resulting triangle weighting automatically has edge sums equal to $1$. The remaining work is to prove this weighting is nonnegative, which is done by rewriting the weight of each ordered triangle as a sum of 'cancelling' pairs of gadget contributions and then solving a maximization program. A symmetrization step reduces the program to ten variables, ramps make the terms nonnegative, and successive monotonicity reductions bring it down to a one-variable bound whose maximum is $3d(1-d)/(1-2d)^2$; requiring this to be at most $1$ fixes $d$ at the stated value.","core_discovery":"The paper's central assertion is Theorem 1.4: if $G$ is a graph on $n$ vertices with minimum degree $\\delta(G) \\ge ((7+\\sqrt{21})/14)n$, then $G$ admits a fractional $K_3$-decomposition—an assignment of nonnegative weights to triangles such that every edge gets total weight $1$. Equivalently, writing $d=(7-\\sqrt{21})/14 \\approx 0.17267$, the theorem holds whenever $\\delta(G) \\ge (1-d)n$, and the constant $d$ is the root of $7d^2-7d+1=0$ in $[0,1/4)$. Combined with the iterative-absorption results cited in the paper, this yields that for every $\\varepsilon>0$, every sufficiently large $K_3$-divisible graph on $n$ vertices with minimum degree at least $((7+\\sqrt{21})/14+\\varepsilon)n$ admits an actual $K_3$-decomposition, and similarly for any 3-chromatic forbidden graph $F$.","pith_inferences":["Beyond the paper: if the two sign errors are repaired, the same delegation-cancellation machinery yields the stated constant, but the paper's own footnote suggests that adding an averaging constraint on the reduced variable could lower the threshold further, toward about $0.813n$.","Beyond the paper: the paper's observation that its weighting fails on near-extremal examples such as clique blow-ups of $C_4$ and independent blow-ups of $K_4$ indicates that closing the gap to the conjectured $3/4$ constant will require a new weighting or delegation rule rather than sharper optimization of the current one.","Beyond the paper: a quick numerical check of the published one- and two-variable programs is a testable way to see whether the claimed bound survives the sign corrections."],"forward_implications":["For every $\\varepsilon>0$, every sufficiently large $K_3$-divisible graph with $\\delta(G)\\ge ((7+\\sqrt{21})/14+\\varepsilon)n$ has a triangle decomposition, the paper's headline progress on the 1970 conjecture.","The same degree threshold controls decompositions into any 3-chromatic graph $F$: every sufficiently large $F$-divisible graph above $(0.82733+\\varepsilon)n$ admits an $F$-decomposition.","The threshold also yields packing corollaries: collections of cycles or 2-regular $n$-vertex graphs pack into hosts of the same high minimum degree as long as their total edge count is bounded away from that of the host.","The constant is not arbitrary: it is the value at which $3d(1-d)/(1-2d)^2=1$, the exact point where the optimization program's upper bound reaches $1$."],"supporting_citations":[{"why":"Supplies the edge-gadget construction whose local weight redistributions form the basis of the paper's weighting.","marker":"[1]"},{"why":"Provides the iterative-absorption result turning a fractional $K_3$-decomposition into a genuine $K_3$-decomposition, yielding Corollary 1.5.","marker":"[2]"},{"why":"Holds the previous hand-verifiable fractional triangle decomposition threshold of $0.9n$ that this paper improves.","marker":"[5]"},{"why":"Extends the fractional-to-integral transfer to all $r$-chromatic graphs, giving the $F$-decomposition corollary.","marker":"[9]"}],"fun_headline_variants":["Fractional triangle decomposition proven at 0.827n","Nash-Williams triangle conjecture: fractional bound 0.827n","Closing gap to Nash-Williams: 0.827n suffices fractionally","Triangle decomposition threshold improved to 0.827n"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the long chain of optimization reductions in Section 5 is correct, but as printed it is not: Claim 5.15.2 miscomputes the derivative of $H(s,t)$ and the final step of Lemma 5.17 asserts $1-5d\\le 0$ for $d\\in[0,1/5]$, which is false, so the claimed upper bound on the program is not rigorously established until those calculations are repaired.","fun_headline_variants_meta":{"raw":{"variants":["Fractional triangle decomposition proven at 0.827n","Nash-Williams triangle conjecture: fractional bound 0.827n","Closing gap to Nash-Williams: 0.827n suffices fractionally","Triangle decomposition threshold improved to 0.827n"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000715,"raw_usage":{"total_tokens":3212,"prompt_tokens":938,"completion_tokens":2274,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":554,"completion_tokens_details":{"reasoning_tokens":2198}},"tokens_in":554,"tokens_out":2274,"duration_ms":16042,"temperature":1.0,"reasoning_tokens":2198,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T05:46:23.015679+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate the published objective $\\widehat{W}_9(a,b)$ on a fine grid over $0\\le a,b\\le d$ for $d=(7-\\sqrt{21})/14$. If any value exceeds $1$, the claimed chain of inequalities $\\mathrm{OPT}(P1)\\le \\mathrm{OPT}(P10)=1$ is false, so the proof as printed cannot establish Theorem 1.4.","supporting_citations":[{"cited_title":"Barber, D","cited_arxiv_id":null,"evidence_quote":"Supplies the edge-gadget construction whose local weight redistributions form the basis of the paper's weighting."},{"cited_title":"Barber, D","cited_arxiv_id":null,"evidence_quote":"Provides the iterative-absorption result turning a fractional $K_3$-decomposition into a genuine $K_3$-decomposition, yielding Corollary 1.5."},{"cited_title":"Dross, Fractional triangle decompositions in graphs with large minimum degree, SIAM Journal on Discrete Mathematics , 30(1), 2015, 36–42","cited_arxiv_id":null,"evidence_quote":"Holds the previous hand-verifiable fractional triangle decomposition threshold of $0.9n$ that this paper improves."},{"cited_title":"Glock, D","cited_arxiv_id":null,"evidence_quote":"Extends the fractional-to-integral transfer to all $r$-chromatic graphs, giving the $F$-decomposition corollary."}],"review_version":1}