{"id":"0efc65d3-c19a-45ed-b401-e414fe31ff3b","arxiv_id":"2607.05896","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"For every ε ≥ 1/2, the maximal anti-Ramsey function χ_S(n, C(n,2)−⌊n^{2−ε}⌋, P_4) exceeds n²/72 for sufficiently large n, establishing the quadratic lower bound in the complementary regime to Li–Ning–Xie's negative result.","lead":"The paper proves that any edge-coloring of a nearly-complete graph forcing every 4-vertex path to be rainbow requires quadratically many colors, settling the remaining open regime of a 1989 problem by Burr, Erdős, Graham, and Sós. A generalist might read it as a clean example of how extremal graph theory resolves decades-old combinatorial conjectures.","discovery_kind":"unclear","skeptic_critique":{"model":"glm-5.2","headline":"Claim 4 in Lemma 1(ii) reverses the adjacency direction: endpoints of e are non-neighbors of v in F (neighbors in H[U]), not neighbors in F. The bound 8n² is numerically correct but derived from the wrong degree sequence.","rationale":"The reader correctly identified the quantitative bounds in Lemma 1(ii) as the delicate part of the argument, but focused on whether the constants (8√n, n/4) are tight rather than on the logical error in Claim 4. The actual issue is more specific: Claim 4 and equations (4)–(6) use d_F(v) (degree in the dense graph G[U]) where they should use d_{H[U]}(v) (degree in the sparse complement H[U]). The paper even writes 'F = H[U]' in equation (4), contradicting its own definition F = G[U]. Taken literally, d_F(v) ≤ 8√n is false for F = G[U], and Σ C(d_F(v),2) would be Θ(n³), collapsing the argument. However, the correction is mechanical: replace d_F(v) with d_{H[U]}(v), and the same chain of inequalities yields Σ C(d_{H[U]}(v),2) ≤ 8n², preserving q > n²/72. The result itself is correct; the proof as written contains a fixable notational/logical error. The verdict remains ACCEPT because the theorem is true and the proof is salvageable with a one-line correction, but the proof as written should not be taken at face value without this fix.","tokens_in":6624,"tokens_out":9372,"duration_ms":645488,"concrete_test":"Re-derive the bound in Claim 4 replacing N_F(v) with N_{H[U]}(v): verify that for (v,i,e) ∈ T with e=xy and f=vw ∈ M_i, the induced matching property gives vx,vy ∉ E(F), hence x,y ∈ N_{H[U]}(v). Then confirm that Σ_{v∈U} C(d_{H[U]}(v),2) ≤ 8n² using d_{H[U]}(v) ≤ 8√n and Σ d_{H[U]}(v) ≤ 2n^{3/2}. If this bound holds (as the arithmetic suggests), the constant n²/72 is valid and the result stands with a corrected proof.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Lemma 1(ii), F is defined as G[U] (a dense graph with |E(F)| > n²/4). Claim 4 states that for (v,i,e) ∈ T with e=xy and f=vw ∈ M_i, 'both x and y belong to N_F(v)'. This is backwards. Since M_i is an induced matching in F, no edge of F connects {v,w} to {x,y}; in particular vx,vy ∉ E(F). So x,y ∈ N_{H[U]}(v), not N_F(v). The injective map (v,i,e) ↦ {x,y} therefore bounds the triple count by C(d_{H[U]}(v), 2), not C(d_F(v), 2). Equation (4) then claims d_F(v) ≤ 8√n by writing 'Since F = H[U]' — but F was defined as G[U], and d_{G[U]}(v) can be as large as |U|-1 ≈ 3n/4. Taken literally, the proof is incorrect: Σ C(d_F(v),2) for a dense graph is Θ(n³), not 8n². However, the fix is straightforward: replace d_F(v) with d_{H[U]}(v) throughout equations (4)–(6). Since v ∈ U gives d_{H[U]}(v) ≤ d_H(v) ≤ 8√n and Σ d_{H[U]}(v) = 2|E(H[U])| ≤ 2|E(H)| ≤ 2n^{3/2}, the same bound Σ C(d_{H[U]}(v),2) ≤ 8n² holds, and the final result q > n²/72 is unchanged. The paper appears to have confused F = G[U] with H[U] in this section, writing 'F = H[U]' in equation (4) despite defining F = G[U] in the lemma statement.","agreement_with_reader":"partial"},"referee_report":{"model":"glm-5.2","summary":"The paper addresses the maximal anti-Ramsey problem of Burr, Erdős, Graham, and Sós for $P_4$. The central result, Theorem 1, establishes that for every fixed $ε ≥ 1/2$ and sufficiently large $n$, $χ_S(n, C(n,2) − ⌊n^{2−ε}⌋, P_4) > n²/72$. This complements the recent negative result of Li–Ning–Xie for $0 < ε < 1/2$, thereby settling the threshold for the problem at $ε = 1/2$. The proof proceeds via Lemma 1, which shows that under the complement sparsity condition $|E(H)| ≤ n^{3/2}$, the edge set of a dense subgraph $F = G[U]$ cannot be partitioned into fewer than $n²/72$ induced matchings, combined with the observation (Claims 2–3) that rainbow-$P_4$ colorings yield induced matching color classes on $F$.","tokens_in":6999,"tokens_out":1055,"duration_ms":161278,"significance":"The result cleanly resolves the complementary regime to Li–Ning–Xie, identifying $ε = 1/2$ as the sharp threshold for the Burr–Erdős–Graham–Sós problem for $P_4$. The proof is short, self-contained, and parameter-free: the constant $1/72$ emerges from Cauchy–Schwarz and the intermediate bounds rather than from any fitted parameter. The structural argument reducing rainbow-$P_4$ colorings to induced matching decompositions is standard but effective. The paper would benefit from addressing the notational issue described below.","major_comments":[{"comment":"Lemma 1(ii), Claim 4 and Eqs. (4)–(6): There is a notational inconsistency that, taken literally, makes the proof incorrect. In the lemma statement, $F$ is defined as $G[U]$. However, in Eq. (4), the manuscript writes 'Since $F = H[U]$,' which contradicts the definition $F = G[U]$. This matters for the degree bound: $d_{G[U]}(v)$ can be as large as $|U|-1 ≈ 3n/4$, so $Σ C(d_F(v), 2)$ for $F = G[U]$ would be $Θ(n³)$, not $8n²$. The fix is straightforward: throughout Eqs. (4)–(6), $d_F(v)$ should be replaced by $d_{H[U]}(v)$. Since $v ∈ U$ implies $d_{H[U]}(v) ≤ d_H(v) ≤ 8√n$ and $Σ_{v∈U} d_{H[U]}(v) = 2|E(H[U])| ≤ 2|E(H)| ≤ 2n^{3/2}$, the bound $Σ C(d_{H[U]}(v), 2) ≤ 8n²$ holds and the final result $q > n²/72$ is unchanged. Additionally, in Claim 4, the assertion that endpoints $x, y$ of $e$ belong to $N_F(v)$ should read $N_{H[U]}(v)$: since $M_i$ is an induced matching in $F = G[U]$, no","section":null}],"minor_comments":[{"comment":"The abstract states 'there is an absolute constant $c > 0$' while Theorem 1 gives the explicit constant $c = 1/72$. Consider stating the explicit constant in the abstract for precision.","section":null},{"comment":"In the proof of Claim 2, the inequality $n − 1 − 8√n > 2$ holds for $n ≥ 36$; specifying this threshold (or simply noting it holds for large $n$) would improve clarity.","section":null},{"comment":"Eq. (5): the identity $Σ_{v∈U} d_F(v) = 2|E(H[U])|$ is correct only after replacing $d_F$ with $d_{H[U]}$; as written with $F = G[U]$, the left side equals $2|E(F)| = 2|E(G[U])|$, which is inconsistent.","section":null},{"comment":"The reference to 'Ruzsa–Szemerédi graphs' in the introduction could cite the original source for completeness.","section":null},{"comment":"Minor typographical issues: the abstract uses $χS$ without consistent subscript formatting; 'Erd˝ os' and 'S´ os' appear with encoding artifacts throughout.","section":null}],"recommendation":"minor_revision","confidential_remarks":"The core mathematical content is correct and the result is a clean complement to Li–Ning–Xie. The only substantive issue is the $F = G[U]$ vs. $F = H[U]$ notational confusion in Lemma 1(ii), which is a presentation error rather than a mathematical one — the fix is mechanical and does not affect the result. I recommend minor revision."},"author_rebuttal":{"model":"glm-5.2","summary":"We thank the referee for the careful reading and for identifying a notational inconsistency in the proof of Lemma 1(ii). The referee is correct that Eqs. (4)–(6) and Claim 4 contain a typo: the degree bounds should be stated in terms of d_{H[U]}(v) rather than d_F(v), since F = G[U] and the relevant sparsity comes from the complement H. We will fix this in the revision. The mathematical argument and the final bound q > n²/72 are unaffected.","responses":[{"response":"The referee is entirely correct, and we are grateful for this careful observation. There is a notational error in Eqs. (4)–(6) and Claim 4. In the lemma statement, F is defined as F = G[U], and this definition is used correctly throughout Claims 1–3 and the statement of part (ii). However, in Eq. (4), the manuscript incorrectly writes 'Since F = H[U],' which contradicts the definition. The intended meaning is that the complement of F is H[U], i.e., H[U] = G[U] = F. The degree bound in Eqs. (4)–(6) should be stated in terms of d_{H[U]}(v), not d_F(v). As the referee notes, d_F(v) = d_{G[U]}(v) can be as large as |U| - 1 ≈ 3n/4, which would make Σ C(d_F(v), 2) = Θ(n³), invalidating the bound 8n². The correct chain of reasoning is: since v ∈ U, we have d_{H[U]}(v) ≤ d_H(v) ≤ 8√n, and Σ_{v∈U} d_{H[U]}(v) = 2|E(H[U])| ≤ 2|E(H)| ≤ 2n^{3/2}. These yield Σ_{v∈U} C(d_{H[U]}(v), 2) ≤ 8n² as written. Similarly, in Claim 4, the assertion that the endpoints x, y of e belong to N_F(v) should read N_{H[U]}(v): since M_i is an induced matching in F = G[U], the absence of edges of F (i.e., G[U]) joining endpoints of f and e means that in the complement H[U], both x and y are neighbors of v. We will replace all occurrences of d_F(v) with d_{H[U]}(v) in Eqs. (4)–(6), correct the statement 'Since F = H[U]' to 'Since the complement of F is H[U],' and replace N_F(v) with N_{H[U]}(v) in Claim 4. The constant 1/72 and all other steps of the proof are unchanged.","revision_made":"yes","referee_comment":"Lemma 1(ii), Claim 4 and Eqs. (4)–(6): There is a notational inconsistency. In the lemma statement, F is defined as G[U], but Eq. (4) writes 'Since F = H[U],' which contradicts the definition F = G[U]. The degree d_{G[U]}(v) can be as large as |U|-1 ≈ 3n/4, so Σ C(d_F(v), 2) for F = G[U] would be Θ(n³), not 8n². The fix: throughout Eqs. (4)–(6), d_F(v) should be replaced by d_{H[U]}(v). Since v ∈ U implies d_{H[U]}(v) ≤ d_H(v) ≤ 8√n and Σ_{v∈U} d_{H[U]}(v) = 2|E(H[U])| ≤ 2|E(H)| ≤ 2n^{3/2}, the bound Σ C(d_{H[U]}(v), 2) ≤ 8n² holds and q > n²/72 is unchanged. Additionally, in Claim 4, the assertion that endpoints x, y of e belong to N_F(v) should read N_{H[U]}(v)."}],"tokens_in":6429,"tokens_out":906,"duration_ms":68233,"standing_objections":[]},"desk_editor":{"model":"glm-5.2","letter":"This paper proves that for every fixed ε ≥ 1/2, χ_S(n, C(n,2) − ⌊n^{2−ε}⌋, P₄) > n²/72 for large n. Combined with the recent negative result of Li–Ning–Xie for ε < 1/2, this identifies ε = 1/2 as the sharp threshold for the 1989 problem. That is the main contribution, and it is a clean resolution of the complementary regime. The proof is short, self-contained, and the constant 1/72 falls out of Cauchy–Schwarz applied to the induced-matching partition — no fitted parameters. Claims 1–3 (density of F, color classes on F are induced matchings) are correct and well-argued. The reduction to induced matchings on a low-complement-degree subgraph is a natural and effective approach. Credit is due for identifying the right structural setup and executing it cleanly. The result is genuinely new and settles half of an open problem from 1989. Now the soft spot. The stress-test note lands: Lemma 1(ii), Claim 4 contains a real notational error. The claim asserts that endpoints x, y of e belong to N_F(v), but the argument actually shows the opposite — since M_i is an induced matching in F, no edge of F joins v to x or y, so x, y are non-neighbors of v in F (equivalently, neighbors in H[U]). The injection should bound the triple count by C(d_{H[U]}(v), 2), not C(d_F(v), 2). Equation (4) then writes “Since F = H[U],” which contradicts the lemma statement defining F = G[U]. Taken literally, d_F(v) for the dense graph F can be Θ(n), making Σ C(d_F(v), 2) = Θ(n³), not 8n². The fix is straightforward: replace d_F(v) with d_{H[U]}(v) throughout (4)–(6). Since v ∈ U gives d_{H[U]}(v) ≤ d_H(v) ≤ 8√n and Σ d_{H[U]}(v) = 2|E(H[U])| ≤ 2n^{3/2}, the bound Σ C(d_{H[U]}(v), 2) ≤ 8n² holds and the final constant 1/72 is unchanged. This is a notational mix-up, not a structural flaw. The argument is sound once corrected. The paper is for extremal graph theorists working on anti-Ramsey or Ruzsa–Szemerédi-type problems. It deserves a serious referee. The referee should ask the author to fix the F/H[U] confusion in Lemma 1(ii), but the theorem and proof strategy are correct.","headline":"Resolves the ε ≥ 1/2 regime of the 1989 Burr–Erdős–Graham–Sós problem for P₄; proof has a fixable notational error in Lemma 1(ii) but the result stands.","tokens_in":7572,"tokens_out":1580,"would_cite":true,"duration_ms":140116,"reading_group":"no","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"glm-5.2","headline":"Quadratic lower bound settles anti-Ramsey problem for P₄ at ε ≥ 1/2","keywords":[],"falsifier":"A construction showing that for some ε ≥ 1/2, one can rainbow-P₄-color a nearly complete graph with o(n²) colors, or an error in the claim that color classes on F must be induced matchings (Claims 2–3), which is the structural step from which the counting argument derives.","tokens_in":6863,"feed_emoji":"🌈","tokens_out":966,"duration_ms":203422,"temperature":0.7,"pith_summary":"The paper resolves the positive half of a 1989 problem of Burr, Erdős, Graham, and Sós about the maximal anti-Ramsey function χ_S(n, e, P₄), which measures the minimum number of colors needed to edge-color an n-vertex graph with at least e edges so that every copy of the 4-vertex path P₄ is rainbow (all edges distinct colors). When the graph is nearly complete — missing only ⌊n^{2−ε}⌋ edges — the question is whether the number of required colors must grow quadratically in n. Li, Ning, and Xie recently showed the answer is no for ε < 1/2. This paper proves the complementary result: for every ε ≥ 1/2, the number of required colors exceeds n²/72 for all sufficiently large n. The proof works by passing to the complement graph H (the missing edges), showing that when |E(H)| ≤ n^{3/2} one can find a large induced subgraph F of G with more than n²/4 edges, and that every rainbow-P₄ coloring of G forces the color classes restricted to F to be induced matchings. A double-counting argument combined with a Cauchy–Schwarz step then forces the number of such induced matchings — and hence the number of colors — above n²/72.","feed_headline":"Quadratic lower bound settles anti-Ramsey problem for P₄ at ε ≥ 1/2","feed_subtitle":"When a nearly complete graph misses at most n^{3/2} edges, every rainbow-P₄ coloring needs Ω(n²) colors, closing the 1989 problem at the ε =","key_machinery":"The central objects are induced matchings — sets of edges no two of which share a vertex and between which no other graph edge connects. The proof establishes that rainbow-P₄ colorings force color classes on a dense subgraph to be induced matchings (Claims 2–3), then uses a double-counting identity relating the number q of induced matchings to the sum of squared matching sizes, bounded above via degree constraints from the complement graph, and bounded below via Cauchy–Schwarz applied to the total edge count of the subgraph.","core_discovery":"The threshold ε = 1/2 is the exact boundary for the maximal anti-Ramsey problem for P₄: below it, quadratic lower bounds fail (by prior work of Li–Ning–Xie); at or above it, any rainbow-P₄ coloring of a nearly complete graph requires Ω(n²) colors. The key mechanism is that when the complement has at most n^{3/2} edges, the color classes on a dense subgraph are forced to be induced matchings, and a counting argument shows there must be at least n²/72 of them.","pith_inferences":[],"forward_implications":["The 1989 problem of Burr, Erdős, Graham, and Sós is now fully resolved for P₄: the answer is positive if and only if ε ≥ 1/2, with the negative regime settled by Li–Ning–Xie and the positive regime settled here.","The constant 1/72 is unlikely to be tight; determining the correct quadratic constant c(ε) remains open.","The technique of reducing rainbow-P₄ colorings to induced-matching decompositions of a dense subgraph may extend to other bipartite host graphs L beyond P₄, particularly those whose rainbow colorings also force induced-matching structure.","The sharpness of the ε = 1/2 threshold — where |E(H)| ≤ n^{3/2} becomes available — suggests a phase transition in the combinatorial structure of nearly complete graphs at this complement-density boundary."],"fun_headline_variants":["Rainbow-P₄ colorings need Ω(n²) colors when complement is sparse enough","Anti-Ramsey threshold for P₄ pinned at ε = 1/2","Color classes forced into induced matchings at the P₄ anti-Ramsey boundary","Quadratic colors required for rainbow P₄ when complement has at most n^{3/2} edges","Burr–Erdős–Graham–Sós P₄ problem resolved at ε ≥ 1/2"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The argument depends on the specific constant 8 in the degree threshold for the vertex set B, and the bound n²/72 emerges from a chain of inequalities that are tight at the ε = 1/2 boundary; any slack in the intermediate degree bounds or edge-count estimates could change the constant, though not the qualitative quadratic growth.","fun_headline_variants_meta":{"raw":{"variants":["Rainbow-P₄ colorings need Ω(n²) colors when complement is sparse enough","Anti-Ramsey threshold for P₄ pinned at ε = 1/2","Color classes forced into induced matchings at the P₄ anti-Ramsey boundary","Quadratic colors required for rainbow P₄ when complement has at most n^{3/2} edges","Burr–Erdős–Graham–Sós P₄ problem resolved at ε ≥ 1/2"]},"model":"glm-5.2","effort":"high","cost_usd":0.0,"raw_usage":{"total_tokens":779,"prompt_tokens":656,"completion_tokens":123,"prompt_tokens_details":null},"tokens_in":656,"tokens_out":123,"duration_ms":45165,"temperature":1.0,"reasoning_tokens":null,"cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-08T21:19:22.217478+00:00","model_set":{"reader":"glm-5.2"},"falsifier":"A construction showing that for some ε ≥ 1/2, one can rainbow-P₄-color a nearly complete graph with o(n²) colors, or an error in the claim that color classes on F must be induced matchings (Claims 2–3), which is the structural step from which the counting argument derives.","supporting_citations":[],"review_version":1}