{"id":"8849457f-9aaf-418f-bbff-c34242648b20","arxiv_id":"2505.01034","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"In red/blue/purple colourings of K_n with no red/purple K_s and no blue/purple K_t, the largest possible number of purple edges is asymptotically RT_s(n,t) when t is linear in n with small slope, and is within a constant (or factor 1/2) of RT_3(n,t) when t is sublinear.","lead":"The paper introduces a Ramsey theory variant where some edges are simultaneously red and blue ('purple'), and studies the maximum number g(n;s,t) of purple edges that can be placed in K_n without creating a red/purple K_s or a blue/purple K_t. It shows that for many parameter ranges, this maximum is asymptotically equal to the classical Ramsey-Turán number RT_s(n,t).","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 4.3(ii) is asserted as an 'easy extraction' from [24, Section 7.7] with no proof or theorem number; Theorem 4.4, and hence the sublinear part of Theorem 1.4(ii), depends on it.","rationale":"I agree with the reader's identification of the weakest assumption. The linear-range results are well-supported: Theorem 1.3's lower bounds are constructive and match known Ramsey-Turán estimates, and Theorem 1.4(i) follows from Brandt, Andrásfai, and Łuczak-Polcyn-Reiher. The sublinear result is the one place where a nontrivial strengthening of an external theorem is asserted without proof. The similar 'quantitative relation easily extracted from [25]' in Theorem 5.2 is lower risk because the exact error only affects a term that is o(n²), and the condition is satisfied for large n under any o(1) error bound. Thus the triangle-free-process extraction is the load-bearing gap. I would keep the paper conditional: accept only after the authors supply the extraction or a precise citation. The computational section is a useful supplement, but shipping code or raw data would be nice rather than necessary for the asymptotic claims.","tokens_in":24663,"tokens_out":29730,"duration_ms":278682,"concrete_test":"Independently re-derive Theorem 4.3(ii) from [24, Section 7.7]: check whether their Proposition/Theorem 7.2 gives the bound uniformly for every m ≤ m*(ε) with failure probability at most e^{-√n}, and identify the constants. Also verify that the E(m) events are nested so that E(m2) implies E(m1). If the extraction fails, recompute the δ in Theorem 4.4 with the strongest bound actually proved in [24]; if no positive δ survives, Theorem 1.4(ii) for t=(√2+ε)√(n log n) should be flagged as unproved.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The paper needs Theorem 4.3(ii): with probability at least 1-e^{-√n}, either E(m) fails or α(G_m) < (√2+10√ε)√(n log n) at m=m*(ε). It says this is 'easily extracted' from [24, Section 7.7] but gives no derivation or specific theorem. Theorem 4.4 uses it with m=m1 to conclude P[α(G_{m1}) ≥ t | E(m2)] = o(1), then colours G_{m1} red and G_{m2}\\G_{m1} purple to get g(n;3,t) ≥ δnt/2 for t=(√2+ε)√(n log n). If the extraction is invalid, that lower-bound result is unsupported. The difficulty is not just the bound itself: Theorem 4.3(ii) must hold at the intermediate time m1, not only for the final graph, with failure probability e^{-√n} after union over many steps. The conditioning step also assumes E(m2) implies E(m1), a nesting property that is asserted but not defined. The main asymptotic equalities, Theorem 1.3 and Theorem 1.4(i), do not rely on this extraction.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript introduces a red/blue/purple Ramsey parameter g(n;s,t), the maximum number of 'purple' edges in a colouring of K_n with no red/purple K_s and no blue/purple K_t, defined for n<R(s,t). The main results are asymptotic formulas comparing g to Ramsey--Turán numbers: Theorem 1.3 gives g(n;s,cn)=(1-o(1))RT_s(n,cn) for small c; Theorem 1.4 gives the triangle case for all linear c except the interval (1/3,4/11) (conditionally on the Andrásfai conjecture inside it), and gives sublinear lower bounds via the triangle-free process; Theorem 1.6 gives a general probabilistic lower bound; Theorem 1.8 gives a conditional Θ(n^2) versus o(n^2) dichotomy. The paper also reports small computational values of a matching variant g_M.","tokens_in":24844,"tokens_out":43379,"duration_ms":384308,"significance":"The paper identifies a natural three-colour Ramsey parameter and shows a close, mostly asymptotic, connection with Ramsey--Turán numbers. The upper bound (1.1) is immediate, so the value of the paper lies in the lower-bound constructions, which use explicit blow-ups, Andrásfai graphs, and the triangle-free process, and which are described in detail with error terms. The results are broad and, if fully supported, constitute a substantial contribution to extremal Ramsey theory. The computational section is a useful supplement. However, the sublinear triangle part of Theorem 1.4(ii) depends on an unproved 'easy extraction' from the Fiz Pontiveros--Griffiths--Morris memoir, which is a load-bearing gap in the current version.","major_comments":[{"comment":"Theorem 4.4, and hence the first part of Theorem 1.4(ii), relies on Theorem 4.3(ii), which is asserted as an 'easy extraction' from [24, Section 7.7] with no theorem number and no derivation. This statement is load-bearing: it must hold at the intermediate deterministic time m1=m*(ε1) with failure probability e^{-√n}, and it is applied after conditioning on E(m2). Please supply the extraction in the text or cite a precise theorem in [24] that implies exactly this form.","section":"Section 4, Theorem 4.3(ii) and Theorem 4.4"},{"comment":"The conditioning step uses the implication that E(m2) implies E(m1) when m2>m1. The events E(m) are never defined in the manuscript. If E(m) is an intersection of good events over the first m steps, this monotonicity is immediate, but it should be stated explicitly; as written, the conditioning argument is incomplete.","section":"Section 4, proof of Theorem 4.4"}],"minor_comments":[{"comment":"The case c=1/2 is not covered by Corollary 2.4, since the Turán graph T_{n,2} has independence number ⌈n/2⌉ ≥ cn for c=1/2. Please add a brief argument for this endpoint, for example using the canonical C5 blow-up with t=n/2−1.","section":"Section 3, proof of Theorem 1.4(i)"},{"comment":"The theorem is stated with a closed interval t ∈ [R^{-1}(s,n) log n, n], but at the lower endpoint t/log n = R^{-1}(s,n) there is no K_s-free graph with α(G) < t/log n under the paper's strict definition of RT_s. Please state the range with a strict inequality or clarify the convention used for RT_s(n,t) at boundary or real parameters.","section":"Section 5.2, Theorem 1.6"},{"comment":"The proof of Theorem 5.3 uses the quantitative form of Theorem 5.2 with an explicit relation between ε, a, and k, which the paper says is 'easily extracted' from [25]. Since this precise statement is used in (5.1)-(5.2), please include the statement or a short derivation.","section":"Section 5.1, Theorem 5.3"},{"comment":"The sentence defining g contains a small wording issue: 'the largest integer such that there exists ... and |P|=g' is circular; it should be rephrased as 'the largest g such that there exists ... with |P|=g'.","section":"Section 1, Definition 1.2"},{"comment":"The sentence 'RTs(n, cn) is known precisely for every c ≥ 1/(s−1)' is inaccurate under the strict definition α(G)<t used in the paper; it should say c > 1/(s−1), consistent with Corollary 2.4.","section":"Section 2.2"}],"recommendation":"major_revision","confidential_remarks":"The paper is likely acceptable after the authors supply the missing extraction from [24] and fix the minor endpoint issues. The main linear-range theorems (Theorem 1.3 and Theorem 1.4(i)) do not depend on the problematic extraction; the sublinear part of Theorem 1.4(ii) does. If the extraction cannot be supplied, the statement of Theorem 1.4(ii) should be weakened accordingly."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The main thing you should know: this paper defines a genuinely new Ramsey-type parameter—g(n;s,t), the maximum number of \"purple\" (double-coloured) edges in a red/blue/purple colouring of K_n with no red/purple K_s and no blue/purple K_t—and proves that for linear t it asymptotically equals the Ramsey-Turán number RT_s(n,t). In other words, the trivial upper bound g ≤ RT is asymptotically tight across a large regime. The novelty is real: Angell's problem and Nagy's density variant are the only predecessors, and the blow-up colouring construction that drives the lower bounds is a nice, reusable tool.\n\nThe main asymptotic equalities hold up. Theorem 1.3 for general s with small c, and Theorem 1.4(i) for triangles, are built on known RT theory (Brandt, Andrásfai, Łuczak–Polcyn–Reiher, Lüders–Reiher), and the proofs are careful with error terms. The paper is honest about its conditional parts: the middle range (1/3, 4/11) is flagged as depending on the Andrásfai conjecture, and Observation 1.5 correctly notes that one cannot expect the constant δ in the sublinear regime to approach 1.\n\nThe soft spot is exactly where the stress-test note puts it: Theorem 4.3(ii). The paper needs a strengthened version of a Fiz Pontiveros–Griffiths–Morris statement—an independence bound on the triangle-free process at the intermediate time m1, with failure probability e^{-√n}—and says it is \"easily extracted\" from FGM Section 7.7, with no proof and no specific theorem number. That statement is load-bearing for Theorem 4.4, which supplies the first part of Theorem 1.4(ii) for t=(1+ε)√(2n log n). The stress-test's second point also lands: the conditioning step asserts E(m2) implies E(m1), a nesting property of the good events that the paper doesn't justify. I don't know whether the extraction is valid—it may be routine for someone inside the triangle-free process—but the paper doesn't demonstrate it, and a reviewer shouldn't have to take it on faith. The linear-range theorems (1.3, 1.4(i)) do not depend on this gap.\n\nThe computational section is a bonus, not a cornerstone; shipping the code and data would be a minor but real improvement. There are also a few \"/suppress\" LaTeX leaks before author names in the text—cosmetic, but they should be cleaned up.\n\nWho this is for: people working in Ramsey–Turán theory and extremal combinatorics. It's a solid within-subfield result, not a field-opening breakthrough, but it introduces a natural variant with clean statements. It deserves a serious referee. My recommendation: send it to peer review, and ask the authors to supply the extraction from FGM (either a proof or a precise citation) before acceptance.","headline":"A genuinely new Ramsey parameter with clean asymptotic results; the load-bearing gap is an unproved 'easy extraction' from FGM that supports only the sublinear part of the triangle theorem.","tokens_in":25448,"tokens_out":6282,"would_cite":false,"duration_ms":55151,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C55","05C35","05D10"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that purple edges — edges coloured both red and blue — can be packed asymptotically exactly as densely as the Ramsey–Turán bound permits.","keywords":["Ramsey numbers","purple edges","Ramsey-Turán numbers","triangle-free process","Andrásfai conjecture","blow-up colouring","independence number","extremal graph theory"],"falsifier":"Simulate the triangle-free process on $n$ vertices up to $m^*(\\varepsilon)=\\left(\\frac12\\sqrt2-\\varepsilon\\right)n^{3/2}\\sqrt{\\log n}$ for small $\\varepsilon>0$; if, while the process is still alive, the empirical probability that $\\alpha(G_m)\\ge(\\sqrt2+10\\sqrt\\varepsilon)\\sqrt{n\\log n}$ exceeds $e^{-\\sqrt n}$ by a non-negligible amount, the strengthened independence claim used for Theorem 4.4 is false.","tokens_in":24393,"feed_emoji":"🟣","tokens_out":12529,"duration_ms":115270,"temperature":0.7,"pith_summary":"A purple edge is one coloured both red and blue. The paper studies $g(n;s,t)$, the largest number of purple edges in a colouring of $K_n$ that contains no red/purple $K_s$ and no blue/purple $K_t$, for $n<R(s,t)$. It shows that in the linear regime $t=cn$ with $c$ small, this number is asymptotically equal to the Ramsey–Turán number $RT_s(n,cn)$ — the largest size of a $K_s$-free graph on $n$ vertices with independence number below $cn$. For triangles the same equality is proved for every constant density outside the interval $(1/3,4/11)$, and inside that interval it is equivalent to the Andrásfai conjecture. The paper also proves that for sublinear $t$ just above $\\sqrt{2n\\log n}$, $g(n;3,t)$ is at least a positive constant fraction of $RT_3(n,t)$, and nearly half of it once $t\\gg\\sqrt{n\\log n}$.","feed_headline":"Purple edges can reach Ramsey–Turán density","feed_subtitle":"Linear forbidden blue cliques: maximum purple-edge count asymptotically matches the Ramsey–Turán bound.","key_machinery":"The central tool is the blow-up colouring. Take a $k$-vertex graph $G$ that is $K_s$-free and has small independence number, replace each vertex by a balanced set, colour the cross-edges that follow $G$'s embedding red, every other cross-edge purple, and all edges inside the parts blue. This produces $\\omega(R\\cup P)=\\omega(G)$, $\\alpha(R)\\le\\lceil n/k\\rceil\\alpha(G)$, and $|P|\\approx e(G)n^2/k^2$. Since $|P|\\le |R\\cup P|\\le RT_s(n,t)$ is trivial, the equality $g\\approx RT_s$ reduces to finding seed graphs with the right edge-to-independence trade-off: regular triangle-free graphs whose independence number equals their degree for $c<1/3$, Andrásfai graphs and their blow-ups for $c>1/3$, K4-free graphs with small independence number for even cliques, and triangle-free-process outputs for the sublinear range.","core_discovery":"For every integer $s\\ge3$ and all sufficiently small $c>0$, $$g(n;s,cn)=(1-o(1))\\,RT_s(n,cn).$$ In words: when the forbidden blue clique is linear in $n$, the maximum number of mutually red-and-blue edges is asymptotically the maximum number of edges in a $K_s$-free graph whose independence number is just below $cn$. The proof matches the trivial upper bound with blow-up colourings seeded by Ramsey–Turán extremal graphs. For $s=3$ the same identity holds unconditionally for all $c\\in(0,1]\\setminus(1/3,4/11)$, and under the Andrásfai conjecture also in the missing interval; for sublinear $t$ the paper establishes weaker but still positive proportions of the Ramsey–Turán maximum.","pith_inferences":["The blow-up colouring is a transfer principle: whenever the extremal $K_s$-free graph with independence number below $t$ is a balanced blow-up of a small seed graph, the equality $g\\approx RT_s$ should follow automatically; this suggests the same mechanism could extend to other forbidden subgraphs whose extremal graphs are blow-up-like.","The sublinear regime between $t=\\sqrt{2n\\log n}$ and $t=4\\sqrt{n\\log n}$ is the part of the triangle story most sensitive to the fine detail of the random process; a direct construction avoiding the extracted independence claim would either close the gap to $\\gamma=\\sqrt2$ or reveal a genuine drop in $g$ near the Ramsey threshold.","The small computational data suggest a testable pattern: when the $n=R(s,t)-1$ critical colouring is unique, a matching of purple edges suffices, whereas in cases with many critical colourings, sparse non-matching purple graphs win; studying the 'swappable-edge' structure of Ramsey-critical graphs could predict where matchings fail."],"forward_implications":["For every fixed $s\\ge3$ and every sufficiently small $c>0$, the maximum number of purple edges in an $(s,cn)$-free red/blue/purple colouring is asymptotically $RT_s(n,cn)$: the trivial upper bound $|P|\\le |R\\cup P|$ is tight.","For triangles, the same asymptotic equality holds unconditionally for all $c\\in(0,1]\\setminus(1/3,4/11)$; if the Andrásfai conjecture is true, it holds for every constant $c\\in(0,1/2]$.","When $t=(1+\\varepsilon)\\sqrt{2n\\log n}$, there are colourings with $g(n;3,t)\\ge \\delta\\,RT_3(n,t)$ for a positive $\\delta$, and when $t\\gg\\sqrt{n\\log n}$, $g(n;3,t)\\ge (1/2-o(1))RT_3(n,t)$.","The lower bound on $t$ is essentially best possible: known Ramsey upper bounds make $g(n;3,t)$ undefined for $t\\le\\gamma\\sqrt{n\\log n}$ with $\\gamma<1/\\sqrt2$, and if the conjectured tight lower bound on $R(3,t)$ is correct the range is essentially complete.","Assuming the conjectured growth of Ramsey numbers, $g(n;s,t)$ is $\\Theta(n^2)$ when $t\\ge R^{-1}((s+1)/2,n)\\log n$ and $o(n^2)$ when $t\\le R^{-1}((s+3)/2,n)$."],"supporting_citations":[{"why":"Supplies triangle-free regular graphs whose independence number equals the degree, the seed graphs for the c<1/3 blow-up lower bound.","marker":"[11]"},{"why":"Provides the triangle-free-process graph and the probability bounds on independence number used in the sublinear theorems.","marker":"[24]"},{"why":"Independent development of the refined triangle-free process used for the same sublinear bounds and the R(3,t) lower bound.","marker":"[8]"},{"why":"Supplies the K4-free graphs with small independence number that seed the even-clique construction.","marker":"[25]"},{"why":"Gives the asymptotic formulas for RT_s(n,cn) that serve as the target upper bounds for Theorem 1.3.","marker":"[34]"},{"why":"Proves the k=2 case of the Andrásfai conjecture, covering densities in [2/5,1/2].","marker":"[3]"},{"why":"Proves the Andrásfai conjecture for k=3, covering densities in [3/8,2/5].","marker":"[31]"},{"why":"Proves the Andrásfai conjecture for k=4, covering densities in [4/11,3/8].","marker":"[33]"},{"why":"Gives the independent-set bound for triangle-free graphs used in Observation 1.5.","marker":"[44]"},{"why":"Gives the phase-transition estimate for RT_s under Conjecture 1.7 used in Theorem 1.8.","marker":"[5]"}],"fun_headline_variants":["Purple edges match Ramsey–Turán bound asymptotically","Linear blue target: purple edge count equals Ramsey–Turán limit","Asymptotic identity: purple edges saturate Ramsey–Turán density","Purple Ramsey: max purple edges reach Ramsey–Turán maximum for linear t","For linear blue cliques, purple edges attain Ramsey–Turán asymptotics"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the triangle-free process, at the stopping time $m^*(\\varepsilon)$, satisfies $\\alpha(G_m) < (\\sqrt2+10\\sqrt\\varepsilon)\\sqrt{n\\log n}$ with probability at least $1-e^{-\\sqrt n}$ whenever the process has not already failed; the paper cites this as 'easily extracted' from a known proof but does not display the extraction.","fun_headline_variants_meta":{"raw":{"variants":["Purple edges match Ramsey–Turán bound asymptotically","Linear blue target: purple edge count equals Ramsey–Turán limit","Asymptotic identity: purple edges saturate Ramsey–Turán density","Purple Ramsey: max purple edges reach Ramsey–Turán maximum for linear t","For linear blue cliques, purple edges attain Ramsey–Turán asymptotics"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000745,"raw_usage":{"total_tokens":3255,"prompt_tokens":813,"completion_tokens":2442,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":429,"completion_tokens_details":{"reasoning_tokens":2347}},"tokens_in":429,"tokens_out":2442,"duration_ms":16387,"temperature":1.0,"reasoning_tokens":2347,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T04:28:43.980098+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate the triangle-free process on $n$ vertices up to $m^*(\\varepsilon)=\\left(\\frac12\\sqrt2-\\varepsilon\\right)n^{3/2}\\sqrt{\\log n}$ for small $\\varepsilon>0$; if, while the process is still alive, the empirical probability that $\\alpha(G_m)\\ge(\\sqrt2+10\\sqrt\\varepsilon)\\sqrt{n\\log n}$ exceeds $e^{-\\sqrt n}$ by a non-negligible amount, the strengthened independence claim used for Theorem 4.4 is false.","supporting_citations":[{"cited_title":"/suppress Luczak, J","cited_arxiv_id":null,"evidence_quote":"Proves the Andrásfai conjecture for k=4, covering densities in [4/11,3/8]."},{"cited_title":"Andr´ asfai","cited_arxiv_id":null,"evidence_quote":"Proves the k=2 case of the Andrásfai conjecture, covering densities in [2/5,1/2]."},{"cited_title":"/suppress Luczak, J","cited_arxiv_id":null,"evidence_quote":"Proves the Andrásfai conjecture for k=3, covering densities in [3/8,2/5]."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the asymptotic formulas for RT_s(n,cn) that serve as the target upper bounds for Theorem 1.3."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies triangle-free regular graphs whose independence number equals the degree, the seed graphs for the c<1/3 blow-up lower bound."},{"cited_title":"Fiz Pontiveros, S","cited_arxiv_id":null,"evidence_quote":"Provides the triangle-free-process graph and the probability bounds on independence number used in the sublinear theorems."},{"cited_title":"Bohman and P","cited_arxiv_id":null,"evidence_quote":"Independent development of the refined triangle-free process used for the same sublinear bounds and the R(3,t) lower bound."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the K4-free graphs with small independence number that seed the even-clique construction."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the independent-set bound for triangle-free graphs used in Observation 1.5."},{"cited_title":"Balogh, P","cited_arxiv_id":null,"evidence_quote":"Gives the phase-transition estimate for RT_s under Conjecture 1.7 used in Theorem 1.8."}],"review_version":1}