{"id":"04971d36-b733-4951-8702-5da8ea8d0063","arxiv_id":"2608.04485","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Exceeding the split-graph spectral threshold forces Omega(m^{(s+t-1)/2}) copies of K_{s,t}^+ and Omega(m^k) copies of C_{2k+1}, both tight up to constant factors.","lead":"This paper proves that m-edge graphs whose spectral radius exceeds the split-graph threshold g_r(m) must contain many copies of two tripartite color-critical graphs: K_{s,t}^+ and odd cycles C_{2k+1}. The lower bounds are of optimal polynomial order, extending spectral existence theorems into the three-chromatic supersaturation regime.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof hinges on Lemma 2.2 (edge-spectral supersaturation-stability), imported from a same-team preprint and not reproduced; if that lemma has hidden hypotheses, the near-bipartite reduction underpinning Theorem 1.5 fails.","rationale":"I read the full argument in good faith and traced the main proof of Theorem 1.5 through the core construction, Lemmas 4.2–4.12, and the two final cases. The internal deductions are coherent: the counting lemmas are dimensionally consistent, the uniqueness of the added edge in K_{s,t}^+ is correctly used to make families of copies disjoint, the sharpness construction in Section 3 has the right orders, and the final contradictions in Case 1 and Case 2 close properly given the stated ε-hierarchy. A brief ambiguity about whether U1' consists only of remaining vertices of U1^(8) is repairable and does not, on a charitable reading, invalidate the matching step. The single most load-bearing assumption is Lemma 2.2, imported from the authors' earlier preprint: the entire reduction of a hypothetical counterexample to a near-complete-bipartite core depends on it, and its proof is not reproduced here. This is exactly the reader's weakest_assumption, so I agree. The right disposition is to keep the reader's CONDITIONAL verdict: conditional on the validity of Lemma 2.2, the main theorems appear to hold, but the paper should supply or verify that stability input before the result is fully accepted.","tokens_in":39814,"tokens_out":30541,"duration_ms":237473,"concrete_test":"Independently re-derive Lemma 2.2(ii) for F=K_{s,t}^+ with t+1≥s≥3, from the assumptions N(F,G)=o(m^{f/2}) and ρ(G)≥√((1−δ)m). Check that the proof yields d(G,K_{U,V})≤εm for arbitrary m-edge G with no isolated vertices, with δ=δ(F,ε) independent of m, and without requiring extra conditions such as minimum degree c√m or s=t. If the proof in [4] uses an unstated hypothesis, the application in Section 4 is unsupported; if the re-derivation goes through, the main caveat is closed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"After passing to the ε0-core H (Definition 4.1), the proof needs a near-bipartite structure to get started: Lemma 4.2 gives only N(K_{s,t}^+,H)=o(h^{(s+t)/2}) and ρ(H)>√h, and Lemma 2.2(ii) (the r=2 case of [4]) is the sole source of the sets U1,U2 with d(H,K_{U1,U2})≤εh. Every subsequent step—Lemmas 4.5–4.12, the Case 1 switch to H*, and the Case 2 deletion of internal edges—assumes that partition. Lemma 2.2 is not proved in this paper; it is stated as a black box from the authors' own preprint arXiv:2603.14964, which is not independently verified. The claimed 'straightforward extension' of Theorem 1.2 to t+1=s is not used in the proof, so the genuinely load-bearing imported result is Lemma 2.2. If Lemma 2.2 has an unstated hypothesis (e.g., a minimum-degree condition or a balancedness assumption), or if its proof in [4] does not cover K_{s,t}^+ with s>t, the central reduction to a near-bipartite core fails.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies edge-spectral supersaturation for two families of 3-chromatic color-critical graphs: the graphs K_{s,t}^+ (complete bipartite K_{s,t} with one additional edge in the part of size s) and odd cycles C_{2k+1}. The main results, Theorem 1.5 and Theorem 1.6, assert that for a large m-edge graph G, the spectral condition rho(G) > g_{s-1}(m) forces Omega(m^{(s+t-1)/2}) copies of K_{s,t}^+, and rho(G) > g_k(m) forces Omega(m^k) copies of C_{2k+1}, with both bounds tight up to constant factors. The proof proceeds by extracting an epsilon-core H from a hypothetical counterexample, applying a stability result to show H is close to a complete bipartite graph, and then counting copies of the forbidden graph through a sequence of structural lemmas. The tightness constructions in Section 3 and Section 6 use split graphs and regular high-girth graphs.","tokens_in":40082,"tokens_out":19869,"duration_ms":169807,"significance":"If the results are correct, they provide the first edge-spectral supersaturation statements in the tripartite, split-threshold regime, extending the existence theorems of Li-Liu-Zhang to optimal polynomial counting bounds and partially resolving Problem 1.4. The paper contains explicit thresholds, matching constructions, and a coherent core-decomposition argument. The main caveat is that the two most powerful tools, Lemmas 2.1 and 2.2, are imported without proof from the authors' companion preprint, so the manuscript is not self-contained at a load-bearing point. I found no free parameters being fit to the target bounds, and the tightness constructions appear correct. The claimed uniqueness of phi in Section 3 is in fact true, although it deserves a proof sketch.","major_comments":[{"comment":"Lemma 2.2 is the unique source of the near-bipartite structure in the proof of Theorem 1.5. It is applied to the epsilon0-core H to produce the partition (U1,U2) with d(H,K_{U1,U2}) <= epsilon h, and every subsequent step - Lemmas 4.5 through 4.12, the construction of H* in Case 1, and the deletion of internal edges in Case 2 - depends on that partition. The lemma is stated as a black box from the authors' own preprint [4] and is not proved in this manuscript. If Lemma 2.2 has unstated hypotheses that fail for K_{s,t}^+ with t+1=s, the central reduction fails. I request that the proof of Lemma 2.2 be reproduced in the paper or in an appendix, or that a published version with a complete proof be cited, with the hypotheses explicitly checked for all parameter ranges in Theorem 1.5.","section":"§2, Lemma 2.2 (used in §4 after (20))"},{"comment":"Lemma 2.1 is also imported from the companion preprint [4] without proof. It is used in Lemma 4.1 to rule out the possibility that too many edges are removed during the epsilon-core iteration, and in Lemma 4.2 to obtain the upper bound rho(H) < sqrt((1+2epsilon)h). These steps are necessary for the core construction and for the subsequent counting argument. The dependency on an unproved companion-paper result should be resolved in the same way as for Lemma 2.2.","section":"§2, Lemma 2.1 (used in Lemmas 4.1 and 4.2)"}],"minor_comments":[{"comment":"The uniqueness of phi satisfying (8) is asserted without proof. The assertion is correct: writing phi = 4sn, the intervals (8s^2(2n-1)(n-1), 8s^2 n(2n+1)] partition the positive integers, so exactly one phi exists for every sufficiently large m. A one-sentence justification would remove any doubt.","section":"§3, inequality (8)"},{"comment":"The displayed equation appears garbled as 'rho2 =rho 2||z||2 2<=...'. Please correct it to the intended algebraic identity, likely rho^2 = rho z^T A(G[U]) z + z^T B B^T z + (1/2)d, so that the subsequent estimates are readable.","section":"§2, Lemma 2.4, equation (4)"},{"comment":"The statement that the methodology of [9] 'can be straightforwardly extended' to the case t+1=s is not proved and is not used later. Either provide a proof or explicitly note that the extension is not needed for the arguments in this paper.","section":"§1, after Theorem 1.2"},{"comment":"The phrase 'the case s=2 corresponds to book graphs' is informal; defining the book graph explicitly would help readers who are not specialists in spectral extremal graph theory.","section":"§1.3, discussion of s=2"},{"comment":"Several key references are arXiv preprints ([4], [7], [8], [9], [10]). If any of these have appeared in refereed journals, the published versions should be cited, especially [4], whose results are load-bearing for the main proof.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The two key lemmas come from the authors' own unpublished preprint [4]. I do not see definitional circularity in the manuscript, but the proof is not self-contained at a load-bearing point. I would recommend that the editor require either a full proof of Lemma 2.2 (and Lemma 2.1) in the manuscript or verification that [4] has been accepted and is publicly available in final form. Absent that, the central claim of Theorem 1.5 cannot be independently checked from the submitted text."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things you should know. First, the main results are genuinely new: Theorems 1.5 and 1.6 give the right polynomial order for K_{s,t}^+ and C_{2k+1} copies above the refined split-graph spectral threshold, and the sharpness constructions are convincing. Second, the proof is not fully self-contained in one important place: the entire reduction to a near-bipartite core uses Lemma 2.2, imported verbatim from the authors' own preprint [4], and that lemma is not proved here. The stress-test note is not manufactured; it is the real load-bearing point. If Lemma 2.2 has hidden hypotheses, or if its proof in [4] does not actually cover K_{s,t}^+ with t+1 >= s, then Theorem 1.5 collapses. I did not find a flaw in the reduction itself, but the authors owe the reader a proof or a precise, verified statement of Lemma 2.2.\n\nThe rest of the paper is mostly solid. The counting arguments in Section 3 and Lemmas 4.5-4.12 are intricate but coherent, and the case analysis in Section 5 seems to check out. The use of Sauer's regular girth lemma for the sharpness construction is clean. The extension to odd cycles and the corollary for intermediate graphs F are natural and follow with standard counting.\n\nSoft spots, in proportion:\n\n1. The claimed uniqueness of phi in Section 3 is false as stated. The interval (8) is very wide for large m, so there are many multiples of 4s satisfying it. This does not hurt the proof, since only existence is needed, but the sentence should be corrected or weakened to \"choose a positive integer divisible by 4s satisfying...\"\n\n2. The \"straightforward extension\" of Theorem 1.2 from [9] to the case t+1=s is asserted without a proof. It is not used later, so the paper would lose nothing by removing the claim or stating it as a remark.\n\n3. The reliance on Lemma 2.2 is the main issue. It is self-citation, not definitional circularity, and I do not see parameter fitting or target bounds assumed as inputs. Still, as the sole source of the near-bipartite partition, it deserves full reproduction here.\n\nVerdict: conditional, but condition is clear. This paper is worth refereeing seriously. I would send it to a competent referee, and I would ask the authors to either prove Lemma 2.2 in the paper or provide an independent verification, and to fix the uniqueness wording. The main theorems are likely correct in substance, and the subfield will cite them if the dependence is cleaned up.","headline":"New and likely correct polynomial-order supersaturation bounds for two chi=3 families, but the proof leans on an unverified same-team stability lemma that should be reproduced before publication.","tokens_in":40583,"tokens_out":1889,"would_cite":true,"duration_ms":19733,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C50","05C35"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that crossing the split-graph spectral threshold forces optimal polynomial many copies of tripartite color-critical graphs $K^+_{s,t}$ and odd cycles.","keywords":["edge-spectral supersaturation","spectral radius","color-critical graph","tripartite graph","odd cycle","split graph","K_{s,t}^+","spectral Turan threshold"],"falsifier":"A direct way to refute Theorem 1.5 would be to produce, for arbitrarily large $m$ and any fixed constant $c>0$, an $m$-edge graph with $\\rho(G)>g_{s-1}(m)$ but fewer than $c m^{(s+t-1)/2}$ copies of $K^+_{s,t}$; no such graph can exist if the theorem is correct. A more targeted check is to test Lemma 2.2 independently by searching for a graph with $\\rho(G)\\approx\\sqrt{m}$, $o(m^{f/2})$ copies of a three-chromatic $F$, and edit distance $\\Omega(m)$ from every complete bipartite graph.","tokens_in":39628,"feed_emoji":"","tokens_out":8378,"duration_ms":71618,"temperature":0.7,"pith_summary":"Edge-spectral extremal graph theory asks what happens when a graph's spectral radius exceeds the largest value allowed while avoiding a prescribed subgraph. This paper proves that for two families of three-chromatic color-critical graphs, crossing the exact split-graph threshold forces not just one forbidden copy but polynomially many. For $K^+_{s,t}$ with $t+1\\ge s\\ge 3$, every sufficiently large $m$-edge graph with $\\rho(G)>g_{s-1}(m)$ contains $\\Omega(m^{(s+t-1)/2})$ copies. For odd cycles, $\\rho(G)>g_k(m)$ forces $\\Omega(m^k)$ copies of $C_{2k+1}$. Both exponents are tight up to constant factors, so the paper turns earlier edge-spectral existence theorems into supersaturation theorems in the harder three-chromatic setting.","feed_headline":"Past a spectral cutoff, forbidden tripartite graphs must multiply","feed_subtitle":"Exceeding the split-graph threshold forces optimal polynomial counts, not just existence, for two 3-color-critical families.","key_machinery":"The central object is the spectral threshold $g_r(m):=\\frac{r-1+\\sqrt{4m-r^2+1}}{2}$, which is the spectral radius of the split graph $S_{r,m}$ (a clique of size $r$ joined to a large independent set, with one extra vertex fixing the residue class). The argument assumes a graph $G$ with $\\rho(G)>g_{s-1}(m)$ and too few copies, extracts an $\\varepsilon$-core subgraph $H$ (no dense subgraphs), and uses the quoted stability lemma to conclude $H$ is $\\varepsilon$-close to a complete bipartite graph $K_{U_1,U_2}$. From then on the Perron eigenvector and the minimal distance partition $(U_1,U_2)$ carry the counting: internal edges inside $U_1$ or $U_2$ are shown to generate many copies of $K^+_{s,t}$ via common neighborhoods, and every other configuration is ruled out by comparing the spectral radius of a modified graph with $g_{s-1}(h)$. The sharpness construction uses a $4s$-regular $C_4$-free graph joined to an independent set.","core_discovery":"The central claim is that the spectral threshold $g_r(m)$ is a genuine supersaturation threshold for these color-critical graphs. Theorem 1.5 states that whenever $t+1\\ge s\\ge3$ and $G$ has $m$ edges with $\\rho(G)>g_{s-1}(m)$, then $N(K^+_{s,t},G)=\\Omega(m^{(s+t-1)/2})$; Theorem 1.6 gives $N(C_{2k+1},G)=\\Omega(m^k)$ under $\\rho(G)>g_k(m)$ for every $k\\ge2$. The paper also constructs graphs showing that neither lower bound can be raised by more than a constant factor. In short, exceeding the threshold that rules out a single copy automatically forces the best possible polynomial surplus of copies.","pith_inferences":["The same $\\varepsilon$-core reduction should transfer to any almost-bipartite graph whose edge-spectral extremal construction is a split graph (for instance fan, friendship, or theta graphs), since the proof itself only uses near-bipartiteness plus counting copies through common neighborhoods.","The sharp constants left open by the two theorems are likely attained by the split constructions themselves, which would make the constant equal to the number of copies inside $S_{s-1,m}$ (or $H_m$) rather than a value coming from an averaging argument.","Because the proof quotes a stability lemma from the authors' earlier work rather than proving it, the robustness of Theorems 1.5 and 1.6 is only as strong as that lemma; a reader wanting full self-containment would need to insert its proof."],"forward_implications":["Corollary 1.7 extends the odd-cycle result to every graph $F$ with $C_{2k+1}\\subseteq F\\subseteq K^+_{k+1,k}$, giving $N(F,G)=\\Omega(m^k)$ under the same spectral condition, again tight up to constants.","The exponents $\\frac{s+t-1}{2}$ and $k$ are best possible: the paper's own constructions satisfy the spectral condition while containing only $\\Theta(m^{\\frac{s+t-1}{2}})$ or $\\Theta(m^k)$ copies, respectively.","Theorem 1.6 resolves the order-of-magnitude part of Problem 1.4 for odd cycles; only the sharp asymptotic constant remains open.","The case $s=2$, which would cover book graphs, is outside the scope of Theorem 1.5 and is not settled by this paper."],"supporting_citations":[{"why":"Supplies Lemmas 2.1 and 2.2, the spectral supersaturation and supersaturation-stability statements that convert a counterexample into a near-bipartite core.","marker":"[4]"},{"why":"Proved the sharp edge-spectral Turan upper bound for K^+_{s,t}-free graphs that identifies the threshold the paper crosses.","marker":"[9]"},{"why":"Proved the sharp odd-cycle spectral threshold via split graphs, the baseline Theorem 1.6 strengthens from existence to supersaturation.","marker":"[6]"},{"why":"Provides the regular high-girth graphs used in the construction showing the copy-count lower bounds are tight up to constants.","marker":"[22]"}],"fun_headline_variants":["Spectral cutoff forces optimal many copies of forbidden tripartite graphs","Crossing spectral threshold multiplies forbidden tripartite graphs","Exceeding spectral bound forces optimal copies of forbidden tripartite graphs","Spectral cutoff triggers polynomial abundance of forbidden tripartite graphs"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof depends on a cited stability lemma (Lemma 2.2 from the authors' previous paper [4]), not proved here, which says that a three-chromatic graph with few forbidden copies and spectral radius near $\\sqrt{m}$ must be nearly bipartite; if that lemma fails, the reduction of a counterexample to a near-bipartite core collapses.","fun_headline_variants_meta":{"raw":{"variants":["Spectral cutoff forces optimal many copies of forbidden tripartite graphs","Crossing spectral threshold multiplies forbidden tripartite graphs","Exceeding spectral bound forces optimal copies of forbidden tripartite graphs","Spectral cutoff triggers polynomial abundance of forbidden tripartite graphs"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.0015,"raw_usage":{"total_tokens":6055,"prompt_tokens":1020,"completion_tokens":5035,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":636,"completion_tokens_details":{"reasoning_tokens":4963}},"tokens_in":636,"tokens_out":5035,"duration_ms":34966,"temperature":1.0,"reasoning_tokens":4963,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T14:39:04.920557+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A direct way to refute Theorem 1.5 would be to produce, for arbitrarily large $m$ and any fixed constant $c>0$, an $m$-edge graph with $\\rho(G)>g_{s-1}(m)$ but fewer than $c m^{(s+t-1)/2}$ copies of $K^+_{s,t}$; no such graph can exist if the theorem is correct. A more targeted check is to test Lemma 2.2 independently by searching for a graph with $\\rho(G)\\approx\\sqrt{m}$, $o(m^{f/2})$ copies of a three-chromatic $F$, and edit distance $\\Omega(m)$ from every complete bipartite graph.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Proved the sharp odd-cycle spectral threshold via split graphs, the baseline Theorem 1.6 strengthens from existence to supersaturation."},{"cited_title":"Sauer, On the existence of regularn-graphs with given girth,J","cited_arxiv_id":null,"evidence_quote":"Provides the regular high-girth graphs used in the construction showing the copy-count lower bounds are tight up to constants."}],"review_version":3}