{"id":"9e2499d5-70bb-437b-bf6d-7d33d3ff2f01","arxiv_id":"2507.09827","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"For any large connected sparse graph G on n vertices, the Ramsey number r(G,tB_k) equals 2n+t-2, extending the tree-book result to all sparse graphs.","lead":"A new proof shows that for large connected sparse graphs, the Ramsey number against a book graph is exactly 2n-1, matching the known tree value. It also handles several disjoint books, giving the exact value 2n+t-2.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 10's exact Ramsey formula depends on sharp, unpublished companion lemmas (Lemma 6 and Lemma 2, plus Lemma 11 from the same preprint [21]); an additive error in Lemma 6's n+k−1 bound would break the greedy step in Case 3.","rationale":"The reader's weakest_assumption identifies Lemma 2 and Lemma 6 from the companion preprints as the main risk, and I agree that this is the dominant vulnerability. The paper's own arguments are standard: induction, Trichotomy, the Bondy-Erdős path-extension lemma, Hall's theorem, and star-book Ramsey. I checked the algebra in Case 3 of Theorem 10 and the constants appear internally consistent at the stated n ≥ 111t^3k^3. The one genuinely load-bearing issue is exactness of the companion bounds, since several inequalities use them with no margin. I additionally note that Lemma 11 from [21] is invoked in Cases 1 and 2 of Theorem 10 and was not explicitly listed by the reader, which is why my agreement is partial rather than full. No verdict change is needed: the conditional verdict already captures this risk.","tokens_in":14893,"tokens_out":57519,"duration_ms":607608,"concrete_test":"Obtain arXiv:2507.03264 and arXiv:2505.04142 and extract the proofs of Lemma 6, Lemma 2, and Lemma 11. Check Lemma 6 in the extremal case G=K1,n−1 and at the edge threshold n(1+1/(24k−12)), verifying that the bound is n+k−1 and not n+k. Check that Lemma 2 yields exactly γ=(q−2)(2s+3ℓ−2)+1 with no unstated minimum-degree or connectivity condition. Then re-run the inequalities in Lemma 7 and Theorem 10 Case 3 with the verified constants. If either lemma has an off-by-one or a hidden restriction, the final blue-degree estimates in Case 3 fail and Theorem 10 would need a weakened statement.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is Theorem 10, proved by induction on t. The induction step invokes Lemma 6 in Lemma 7 and again in Cases 2 and 3 of Theorem 10, always using the exact bound r(G,K1,k) ≤ n+k−1. In the final Case 3 argument, the available blue degree of the failing vertex z is bounded below by n+k−1+|(t−1)Bk|; if Lemma 6's true bound were n+k, this margin would vanish and the contradiction would not follow. Lemma 2 supplies the structural trichotomy in Case 3 of both Theorem 8 and Theorem 10; its γ=(q−2)(2s+3ℓ−2)+1 controls every later inequality involving n−γ. Additionally, Cases 1 and 2 of Theorem 10 use Lemma 11, also from [21], to force a blue tK2. All three lemmas are stated in the authors' own unrefereed companion preprints, so the main theorem is exactly as secure as those preprints. I found no internal contradiction in the present text; the only minor internal gap I noticed is Lemma 7's unproved assertion ℓ≥3, which fails for trees (where G0 becomes K1), but this is patchable by classical tree results and does not threaten the central claim. The dominant risk remains the exact companion-lemma bounds with no slack.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies Ramsey numbers of connected sparse graphs versus disjoint books. Let B_k denote the book K_2+K_k and tB_k its disjoint union. The main result, Theorem 10, asserts that for positive integers k,t and n >= 111 t^3 k^3, every connected graph G on n vertices with at most n(1 + 1/(127 t^2 k^2 + 79 t^2 k)) edges satisfies r(G, tB_k) = 2n + t - 2. Theorem 8 establishes the single-book case t=1 with n >= 34 k^3 and the edge bound n(1 + 1/(119 k^2 + 62 k)), and Theorem 9 proves the star case r(K_{1,n-1}, tB_k) = 2n + t - 2 for n >= 3tk + 3t - 5. The lower bound is Burr's goodness bound; the upper bound is proved by induction on t, using a trichotomy lemma for sparse graphs to split into cases of a long suspended path, many end-edges, or a core of bounded size, and then either extending a red copy of a subgraph of G or forcing a blue book. The proofs are built on three key lemmas that are quoted from the authors' own companion preprints: Lemma 2 (Trichotomy), Lemma 6 (star Ramsey bound), and Lemma 11 (tK_2-goodness).","tokens_in":15199,"tokens_out":15937,"duration_ms":160254,"significance":"If the companion lemmas are correct, this is a substantial extension of the classical Erdős-Faudree-Rousseau-Schelp theorem that large trees are B_k-good, and it extends the Luo-Peng result on trees versus tK_k to sparse graphs versus disjoint books. The paper gives explicit polynomial lower bounds on n and explicit constants in the edge condition, and it is honest about the trade-off between n and e(G). The proofs are detailed and use appropriate standard tools: the Bondy-Erdős path lemma, Hall's theorem, the Andrásfai-Erdős-Sós theorem, and Burr's lower bound. I did not find an internal contradiction in the present text. The main weakness is verification: the exact numerical bounds in Lemma 2, Lemma 6, and Lemma 11 come from unreviewed preprints by the same authors, and the main theorem uses those bounds with no slack.","major_comments":[{"comment":"The main theorems depend on three lemmas quoted from the authors' own unreviewed preprints [16] and [21], and the dependence is exact rather than asymptotic. Lemma 6 is invoked in Lemma 7 and again in Cases 2 and 3 of Theorem 10 with the precise bound r(G, K_{1,k}) <= n + k - 1; in the final greedy step of Theorem 10, the blue-degree lower bound is n + k - 1 + |(t-1)B_k|, so if the true bound were n + k the contradiction would disappear. Lemma 2 supplies the structural trichotomy and the parameter gamma = (q-2)(2s + 3l - 2) + 1 that controls every later inequality in Case 3 of Theorems 8 and 10. Lemma 11 is used in Cases 1 and 2 of Theorem 10 to force a blue tK_2. To make the paper self-contained and the main theorem verifiable, the proofs of these three lemmas should be included in this manuscript (for example, in an appendix) or the lemmas should be replaced by published versions with identical hypotheses.","section":"Sections 2, 3, and 5 (Lemmas 2, 6, and 11)"},{"comment":"The proof asserts 'since K_N contains no red G_0 and G_0 is connected, we must have l >= 3'. This is false when G is a tree: after deleting all degree-1 vertices recursively and shortening suspended paths, G_0 is a single vertex (l = 1). The subsequent estimates 2(l-1)/l >= 1 and the application of Lemma 5 to r(G_0, B_k) do not apply as written. The tree case is presumably covered by the classical theorem of Erdős, Faudree, Rousseau, and Schelp (Theorem 7 in the paper), but the proof of Lemma 7 should either handle this case explicitly or restrict the argument to the situation in which G_0 has at least two vertices.","section":"Section 3, Lemma 7"}],"minor_comments":[{"comment":"The graph G' is used before it is defined: the sentence 'let G_1 be a graph obtained from G' by deleting the vertices of degree 1' introduces G' only later. Please reorder the definitions for clarity.","section":"Section 3, Lemma 7"},{"comment":"Substituting c = 49 into g(k,c) = (2c+21)k^2 + (c+25/2)k gives 119k^2 + 61.5k, whereas Theorem 8 uses 119k^2 + 62k. The constants in the concluding remark should be reconciled with the statements of Theorems 8 and 10.","section":"Section 6, Concluding Remark"},{"comment":"In Cases 1 and 2, the use of Lemmas 10 and 11 to force a blue tK_2 requires checking the hypothesis e(G) <= n + n^2/(4t-5) - 2. This follows from n >= 111t^3k^3 and the given edge bound, but the verification is not shown and should be included.","section":"Section 5, Theorem 10"}],"recommendation":"major_revision","confidential_remarks":"The central obstacle is the reliance on two companion preprints by the same authors, with exact bounds that are used with no slack. If the editors require self-contained proofs for published papers, the manuscript should not be accepted until those lemmas are either proved in an appendix or published and publicly available. The in-paper proof of the main theorem is otherwise structured and plausible, and the tree gap in Lemma 7 is patchable. The paper fits the journal's scope and would be a solid contribution once the external dependence is resolved."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This is a solid, workmanlike contribution, and the main formula is genuinely new: r(G,tB_k)=2n+t-2 for every large connected sparse G, extending the 1988 tree-book theorem to multiple books and to graphs that are far from trees. Theorem 9, the star case, is also new and is proved with a clean induction using the Andrásfai–Erdős–Sós bound. The paper is clearly organized, and Lemma 5's self-contained bound on r(G,B_k) is a useful tool in its own right.\n\nThe soft spot is not in the internal logic—I saw no contradiction and the arithmetic checks are consistent—but in the load-bearing dependence on two companion preprints from the same group: Lemma 2 (trichotomy) and Lemma 6 (star Ramsey bound), plus Lemma 11 for tK2 in Theorem 10. The bounds are used exactly, with no slack. In particular, Lemma 6's r(G,K_{1,k}) ≤ n+k−1 is invoked in Lemma 7 and again in Cases 2 and 3 of Theorem 10; if the true constant were n+k, the final greedy arguments would lose their margin and the contradiction would not follow. That is a real verification gap, and the paper cannot be fully trusted until those preprints are checked. The hand-tuned thresholds make the result true but ugly; the concluding remark is honest that they are not tight.\n\nOne small internal gap: Lemma 7 asserts G0 has at least 3 vertices (ℓ≥3), which fails for trees, but the tree case is already covered by the classic theorem, so this is patchable and not a threat to the main claim.\n\nWho should read this: anyone working on Ramsey goodness or book Ramsey numbers. It deserves a serious referee, but the referee must be asked to verify Lemma 2, Lemma 6, and Lemma 11, or to have the authors fold those proofs in. I would not desk-reject it.","headline":"The exact Ramsey formula for sparse graphs versus disjoint books is new and the proof is coherent, but the main theorem's fate hangs on two unrefereed companion preprints used at their exact stated bounds.","tokens_in":15689,"tokens_out":2081,"would_cite":true,"duration_ms":22560,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C55","05C35"],"pacs":[],"model":"deepseek-v4-flash","headline":"For all sufficiently large connected sparse graphs $G$, the Ramsey number against $t$ disjoint books $B_k$ is exactly $2n+t-2$.","keywords":["Ramsey number","sparse graph","book graph","Ramsey goodness","disjoint books","suspended path","structural trichotomy"],"falsifier":"Search, for the smallest admissible $k$ and $t$, for a connected graph $G$ with $n\\ge 111t^3k^3$ vertices and at most $n(1+1/(127t^2k^2+79t^2k))$ edges such that some red-blue coloring of $K_{2n+t-2}$ contains neither a red $G$ nor a blue $tB_k$; any such pair $(G,\\text{coloring})$ would disprove the exact formula, and for moderate $n$ such a search is feasible by SAT-based Ramsey verification.","tokens_in":14702,"feed_emoji":"📚","tokens_out":16882,"duration_ms":181864,"temperature":0.7,"pith_summary":"This paper proves exact Ramsey numbers for all sufficiently large connected sparse graphs against disjoint books. A book $B_k$ is $k$ triangles sharing a common edge, and $tB_k$ is $t$ disjoint copies. The main theorem states that if $G$ is connected with $n\\ge 111t^3k^3$ vertices and at most $n(1+1/(127t^2k^2+79t^2k))$ edges, then $r(G,tB_k)=2n+t-2$. Since the general lower bound from Theorem 1 gives exactly $2n+t-2$ for $H=tB_k$, the theorem says every such graph is $tB_k$-good. It thereby extends the 1988 tree-versus-book result to arbitrary sparse connected graphs and to any number of books, and its proof also settles the single-book case and the star case.","feed_headline":"All large sparse graphs share one book-Ramsey number","feed_subtitle":"At 2n+t−2, every red-blue coloring forces a red sparse graph or t blue books.","key_machinery":"The central object is the book $B_k=K_2+\\overline{K_k}$ (equivalently, $k$ triangles sharing a common edge), and the load-bearing tool is the companion Trichotomy Lemma. It asserts that a connected sparse graph either has a suspended path of a specified length, or has a matching of a specified number of end-edges, or has a small set of vertices of degree at least $2$ together with a vertex adjacent to many leaves. Each branch is settled by a different mechanism: the path-extension lemma to lengthen a suspended path, a matching lemma (Lemma 4) combined with the star bound $r(G,K_{1,k})\\le n+k-1$ to attach end-edges, and an embedded star $K_{1,n-1}$ whose center absorbs all remaining leaves greedily. A self-contained bound $r(G,B_k)\\le n+2km-2m/n$ controls the size of the graph that survives after pruning and is combined with the star bound to obtain the intermediate lemma $r(G,B_k)\\le 2n+k-2$.","core_discovery":"The central claim is Theorem 10: for positive integers $k,t$ and $n\\ge 111t^3k^3$, every connected graph $G$ on $n$ vertices with $e(G)\\le n(1+1/(127t^2k^2+79t^2k))$ satisfies $r(G,tB_k)=2n+t-2$, where $B_k=K_2+\\overline{K_k}$ is the book graph on $k+2$ vertices. Because $\\chi(tB_k)=3$ and its chromatic surplus is $t$, the lower bound of Theorem 1 gives $(n-1)(3-1)+t=2n+t-2$; the theorem is therefore the statement that all such sparse connected graphs are $tB_k$-good. The upper-bound proof splits into three structural cases — a long suspended path, a large matching of end-edges, or a small core with a vertex adjacent to many leaves — and in every red-blue coloring of $K_{2n+t-2}$ one of the cases forces either a red copy of $G$ or a blue $tB_k$. Along the way the paper proves the single-book theorem $r(G,B_k)=2n-1$ and the star theorem $r(K_{1,n-1},tB_k)=2n+t-2$.","pith_inferences":["Because the value $2n+t-2$ never involves $k$, the page size of the book only enters through the hypotheses; a natural test is whether the same exact value persists for much larger edge allowances or for other 3-chromatic graphs with chromatic surplus $t$.","The edge budget is only about $n$ plus $n/(127t^2k^2+79t^2k)$ edges, so the class covered has average degree just above 2; the result implies book-goodness is a low-average-degree phenomenon rather than a tree-only or cycle-only phenomenon.","The concluding remark exposes a trade-off between the lower bound on $n$ and the upper bound on $e(G)$; an immediate next step, even within the same proof framework, is to determine the actual trade-off curve and whether constants such as $111$, $127$, and $79$ can be substantially reduced."],"forward_implications":["If the main theorem is correct, then for $n\\ge 34k^3$ every connected graph with at most $n(1+1/(119k^2+62k))$ edges satisfies $r(G,B_k)=2n-1$: all such graphs are $B_k$-good.","For stars, $r(K_{1,n-1},tB_k)=2n+t-2$ whenever $n\\ge 3tk+3t-5$, so stars of any size order are $tB_k$-good.","For the full range $n\\ge 111t^3k^3$, the exact Ramsey number of a sparse connected graph against $tB_k$ depends only on $n$ and $t$, not on the graph's internal structure or on the page size $k$ beyond the hypotheses.","Consequently equality holds in the general lower bound $r(G,H)\\ge(n-1)(\\chi(H)-1)+s$ for every connected sparse graph $G$ in the stated range, i.e. all these graphs are $tB_k$-good in the sense of Ramsey goodness."],"supporting_citations":[{"why":"Supplies Lemma 2, the Trichotomy Lemma that structures every case-split in the proofs of Theorems 8 and 10, and Lemma 11 for the Ramsey number against $tK_2$.","marker":"[21]"},{"why":"Supplies Lemma 6, the bound $r(G,K_{1,k})\\le n+k-1$, used to force a blue star inside a blue-free coloring and to embed red stars in the final cases.","marker":"[16]"},{"why":"Supplies Theorem 1, the lower bound $r(G,H)\\ge(n-1)(\\chi(H)-1)+s$, together with the notion of $H$-goodness that the paper verifies for sparse graphs and books.","marker":"[3]"},{"why":"Establishes the tree-versus-book theorem $r(T_n,B_k)=2n-1$ that this paper extends to all sparse connected graphs and to multiple disjoint books.","marker":"[10]"},{"why":"Supplies Lemma 3, the path-extension lemma used in the suspended-path cases to force either a blue $B_k$ or a longer red path.","marker":"[2]"},{"why":"Supplies Lemma 4, the matching lemma used in the end-edge cases to conclude either a red matching covering prescribed vertices or a blue complete bipartite structure.","marker":"[14]"},{"why":"Supplies Theorem 6, the star-versus-book value $r(K_{1,n-1},B_k)=2n-1$, which is the base of Theorem 9 and the star-embedding tool in the core case.","marker":"[19]"},{"why":"Supplies Lemma 9, the multi-copy upper bound $r(G,tH)\\le r(G,H)+(t-1)|H|$, used to lift single-book bounds to $t$-book upper bounds.","marker":"[6]"},{"why":"Supplies Lemma 10, the bound $r(G,2K_2)=n+1$, used together with Lemma 11 to find a blue $tK_2$ when extending paths or matching edges.","marker":"[9]"}],"fun_headline_variants":["Sparse graphs share one Ramsey number with disjoint books","For sparse graphs, r(G,tB_k)=2n+t−2 holds uniformly","Book-Ramsey numbers collapsed to 2n+t−2 for sparse graphs","Every sparse connected graph matches disjoint-book Ramsey bound"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The upper-bound argument stands on two companion lemmas taken from the authors' preprints and not reproved here: the Trichotomy Lemma and the bound $r(G,K_{1,k})\\le n+k-1$; if either companion result has an unstated condition or a hidden gap, the proof of Theorem 10 fails even if the theorem is true.","fun_headline_variants_meta":{"raw":{"variants":["Sparse graphs share one Ramsey number with disjoint books","For sparse graphs, r(G,tB_k)=2n+t−2 holds uniformly","Book-Ramsey numbers collapsed to 2n+t−2 for sparse graphs","Every sparse connected graph matches disjoint-book Ramsey bound"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000639,"raw_usage":{"total_tokens":2944,"prompt_tokens":947,"completion_tokens":1997,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":563,"completion_tokens_details":{"reasoning_tokens":1922}},"tokens_in":563,"tokens_out":1997,"duration_ms":16764,"temperature":1.0,"reasoning_tokens":1922,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T17:47:10.292593+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Search, for the smallest admissible $k$ and $t$, for a connected graph $G$ with $n\\ge 111t^3k^3$ vertices and at most $n(1+1/(127t^2k^2+79t^2k))$ edges such that some red-blue coloring of $K_{2n+t-2}$ contains neither a red $G$ nor a blue $tB_k$; any such pair $(G,\\text{coloring})$ would disprove the exact formula, and for moderate $n$ such a search is feasible by SAT-based Ramsey verification.","supporting_citations":[{"cited_title":"Bondy and P","cited_arxiv_id":null,"evidence_quote":"Supplies Lemma 3, the path-extension lemma used in the suspended-path cases to force either a blue $B_k$ or a longer red path."}],"review_version":1}