{"id":"2d923ed3-b084-49e0-9399-ab4c08f0b769","arxiv_id":"2502.09830","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For cycles and balanced complete multipartite graphs, every forest of copies of F is forced in large Ramsey graphs, but infinitely many graphs F admit Ramsey graphs that avoid all non-trivial such forests.","lead":"This paper studies which small configurations of a target graph F must appear in every large Ramsey graph, meaning any edge-coloring produces a monochromatic F. It shows certain configurations are forced for cycles and cliques, while for infinitely many other graphs they are avoidable.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.1, the ordered linear hypergraph girth Ramsey theorem, is stated without proof or explicit citation, and Theorems 1.4 and 1.6 rely on it via Corollary 3.4; if it is not established in [10], the negative results lack foundation.","rationale":"The strongest claim is a dichotomy: for some F every forest of copies is unavoidable (Theorem 1.3), while for infinitely many F some forest of copies is avoidable (Theorem 1.4). The negative side and the density result Theorem 1.6 both flow from Corollary 3.4, whose proof is entirely conditional on Theorem 3.1. Since Theorem 3.1 is stated with neither proof nor a pointer to a specific result in the cited preprint [10], its correctness is the least secure load-bearing premise. Other potential gaps I examined are less serious: the terse step in Claim 4.5 can be justified by the 4-connectivity of F (if the shared intersection q has size at most one, then q would be a cut vertex of the copy of F, contradicting 4-connectivity), and the use of property (ii) in Theorem 1.6 can be bypassed because any copy of F using only forward or only backward edges automatically contains a monotone copy of a longest path, since the path's edges are all directed forward or backward. These observations leave Theorem 3.1 as the decisive unresolved dependency. The reader's conditional verdict is therefore appropriate: if Theorem 3.1 is verified, the paper's main results stand; if not, the negative results and the dichotomy are unproven. My read does not change the verdict, so I leave it unchanged.","tokens_in":14236,"tokens_out":33539,"duration_ms":298103,"concrete_test":"Inspect the proof in Reiher–Rödl, arXiv:2308.15589, to determine whether it proves the ordered linear hypergraph version of the girth Ramsey theorem, i.e., the statement of Theorem 3.1 here. If it appears there, even as a lemma, the concern is resolved. If not, attempt to adapt the proof of Theorem 1.2 to k-uniform linear hypergraphs with a total order; if the adaptation succeeds without new ideas, note that in a revision; if it fails or requires a substantive new argument, the paper must either prove Theorem 3.1 or restrict Theorems 1.4 and 1.6 to cases where the graph proof directly transfers.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's dichotomy depends on the negative direction, Theorem 1.4, whose proof in §4.2 applies Corollary 3.4, itself derived from Theorem 3.1 in §3. Theorem 3.1 is an ordered, linear k-uniform hypergraph extension of the graph girth Ramsey theorem (Theorem 1.2), but it is stated only with 'can be stated as follows' and no proof or explicit reference to [10]. The graph version is cited to [10], yet the hypergraph version is not. This is not a routine restatement: it adds hypergraph uniformity, linearity, and a total order, and it is used in Corollary 3.4(ii) to control induced copies and in the forest-of-copies properties. If Theorem 3.1 is not already proved in [10], Theorems 1.4 and 1.6 are unsupported. The cycle case of Theorem 1.3 also omits the induction details, but that is a minor gap compared with an unproven black box.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies which subgraphs are unavoidable in Ramsey graphs for a given graph F, both in the induced and the non-necessarily-induced edge-Ramsey senses. It builds on the girth Ramsey theorem of Reiher and Rödl (Theorem 1.2) and states an ordered, linear hypergraph version of that theorem (Theorem 3.1). The main new results are: Theorem 1.3, showing that for every cycle or balanced complete multipartite graph F and every forest of copies of F, every Ramsey graph for F with sufficiently many colours contains a subgraph isomorphic to the union of that forest; Theorem 1.4, constructing infinitely many graphs F for which there are Ramsey graphs in which all copies of F are induced and any two copies sharing an edge also share a third vertex non-adjacent to that edge; Theorem 1.5, producing Ramsey graphs that are locally not 2-Ramsey for any graph containing a cycle; and Theorem 1.6, giving bipartite graphs F for which there are Ramsey graphs containing F-free subgraphs with positive density. The paper also derives Theorem 1.8, that every graph containing a cycle is Ramsey infinite, as a consequence of Theorem 1.5. The proofs of the positive results and of the auxiliary propositions about 2-density are given in detail, while the proof of Theorem 1.4 and part of the proof of Theorem 1.6 rely essentially on Theorem 3.1.","tokens_in":14463,"tokens_out":7714,"duration_ms":73936,"significance":"If the standing theorems are accepted, the paper makes a solid contribution to Ramsey theory. The dichotomy expressed by Theorems 1.3 and 1.4 is natural and answers a converse question to the girth Ramsey theorem in a clean way. The paper contains several fully worked technical ingredients, including the set-mapping arguments in Lemmas 2.2 and 2.3, the separability argument in Lemma 3.3, the explicit construction of infinitely many (e,v)-inseparable 3-uniform hypergraphs in Proposition 4.1, and the detailed density calculations in Propositions 5.1 through 5.4. Corollary 5.5, which says that forests of copies of a cyclic graph are not 2-Ramsey, is an elegant and useful observation. The main caveat is that the ordered hypergraph girth Ramsey theorem, Theorem 3.1, is used as a black box without proof or explicit citation; this theorem is load-bearing for the negative results in Theorems 1.4 and 1.6.","major_comments":[{"comment":"Theorem 3.1 is stated without proof and without an explicit citation. The sentence introducing it says the variant 'can be stated as follows,' and the reference [10] is given only for the graph version, Theorem 1.2. The ordered hypergraph statement is not a notational restatement of Theorem 1.2: it concerns k-uniform linear hypergraphs, a total order on vertices, ordered induced copies, and a parameter n for the size of a family of copies. The proof of Corollary 3.4 uses Theorem 3.1 for all three properties (i)--(iii), and Corollary 3.4 is then used in the proofs of Theorem 1.4 (Section 4.2) and Theorem 1.6 (Section 6). Please provide a proof of Theorem 3.1, or give an exact pointer to the statement in [10] and derive the present formulation from it.","section":"Section 3, Theorem 3.1"}],"minor_comments":[{"comment":"The cycle case of Theorem 1.3 is not fully written out: the final sentence says that the proof imitates the complete multipartite case and omits the details. Since Lemma 2.3 is proved in full, this is a presentation gap rather than a fundamental one, but a short paragraph showing the induction step would make the proof self-contained.","section":"Section 2.2, final paragraph"},{"comment":"The proof of the minimum degree claim says 'one can check' and lists four intervals for I(x); expanding the verification would improve readability, especially because the interval definitions and the counting of pairs {a,b} are central to the degree bound.","section":"Section 4.1, Claim 4.2"},{"comment":"The sentence 'since F* is 4-connected, we arrive at V(F*)=V(S_t)' is quite terse; a few explanatory sentences about how the 4-connectivity rules out vertices of F* outside S_t would help the reader follow the minimality argument.","section":"Section 4.2, Claim 4.5"},{"comment":"The notation F is used both for the forest of copies and for the graph F = union F; although the context is usually clear, a distinct symbol for the forest would remove the ambiguity, especially in the proof of Proposition 5.1.","section":"Section 5.1, Proposition 5.1"}],"recommendation":"major_revision","confidential_remarks":"The paper relies heavily on a preprint by two of the same authors, and the crucial ordered hypergraph version is neither proved nor explicitly located in that preprint. The editor may want to ask the authors to confirm that Theorem 3.1 appears in [10] in essentially this form, or to include a proof in the present paper. If that theorem is available, the rest of the paper appears to be a coherent and valuable contribution within the scope of the journal."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This paper gives a genuinely new answer to a natural structural question: when must every Ramsey graph for F contain every forest of copies of F? Theorem 1.3 covers cycles and balanced complete multipartite graphs; Theorem 1.4 shows the opposite for infinitely many dense, 4-connected graphs. That contrast is the real contribution and it is not just a restatement of the girth Ramsey theorem.\n\nWhat is new and good: Theorems 1.3, 1.4, 1.5, 1.6 are original. The proofs of the main lemmas (2.2, 2.3, 3.3, 5.1-5.4) are given in detail and look sound. The 2-density arguments in Section 5 are particularly clean. The paper is honest about its limits—Theorem 1.6 is explicitly noted to give less than half the edges, and the authors raise Question 1.7. The reliance on the two-author girth Ramsey theorem is legitimate; the issue is not self-citation but whether the hypergraph variant is actually available.\n\nThe soft spot: Theorem 3.1 is stated as \"can be stated as follows\" with no proof and no explicit reference. It adds k-uniformity, linearity, and a total order to the graph version, and Corollary 3.4 uses it to get control on induced copies. Theorems 1.4 and 1.6 rest on that. This is not a routine restatement, so it is a real gap in the paper as written. It is fixable—the authors either prove it or point to where in [10] it appears—but it is load-bearing. The cycle case of Theorem 1.3 omits details, but that is minor given Lemma 2.3 is proven and the reduction mirrors the multipartite case.\n\nSummary: this is a paper for Ramsey theory and extremal graph theory researchers. It deserves serious peer review. I would send it to referees and ask for the Theorem 3.1 gap to be resolved before publication.","headline":"A sharp dichotomy for unavoidable forests of copies in Ramsey graphs, built on the girth Ramsey theorem; the load-bearing ordered hypergraph variant needs a proof or a citation.","tokens_in":15029,"tokens_out":2489,"would_cite":true,"duration_ms":21675,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05D10","05C55"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper establishes a precise dichotomy: for cycles and balanced complete multipartite graphs every forest of copies is forced in every Ramsey graph with sufficiently many colours, while for infinitely many other graphs there exist…","keywords":["Ramsey graphs","forests of copies","girth Ramsey theorem","ordered linear hypergraphs","Ramsey infinite","2-density","cycles","complete multipartite graphs"],"falsifier":"Two checks would settle the claims: (1) inspect whether the proof of the girth Ramsey theorem for graphs in [10] extends verbatim to ordered linear hypergraphs as Theorem 3.1 requires; (2) for $n=16$ and $r=2$, build the graph $F$ from Proposition 4.1 and the Ramsey graph $G$ from Corollary 3.4, then look for two induced copies of $F$ in $G$ that share an edge without a common third vertex non-adjacent to that edge, or for any non-induced copy of $F$—either would refute Theorem 1.4.","tokens_in":14032,"feed_emoji":"🌲","tokens_out":14514,"duration_ms":117139,"temperature":0.7,"pith_summary":"The paper asks which local configurations of copies of a graph $F$ must appear in every large Ramsey graph for $F$. Using the girth Ramsey theorem, it proves that for cycles and for balanced complete multipartite graphs, every 'forest of copies'—a collection of copies glued along a single vertex or a shared edge—is unavoidable: any non-induced Ramsey graph with sufficiently many colours must contain the entire forest as a subgraph. In the opposite direction, it constructs infinitely many graphs $F$ for which this fails in the strongest way: for every number of colours there is a Ramsey graph in which all copies of $F$ are induced, and any two copies sharing an edge also share a third vertex that is adjacent to neither endpoint, so no genuinely non-trivial forest of copies exists. The paper also shows that Ramsey graphs can be made locally non-Ramsey on any fixed number of vertices, and that certain bipartite $F$ admit Ramsey graphs whose every subgraph contains an $F$-free subgraph with a positive fraction of the edges. Together these results delineate exactly how much forest-like structure the girth Ramsey theorem forces, and when it can be completely suppressed.","feed_headline":"Forests of copies: forced for cycles, avoidable for infinitely many","feed_subtitle":"Cycles and multipartite graphs force every forest of copies; infinitely many graphs allow Ramsey graphs with none.","key_machinery":"The engine is the girth Ramsey theorem (Theorem 1.2): for any graph $F$ and integers $r$ and $\\ell$ there is a Ramsey graph $G \\to (F)_r$ with a system of copies in which every subfamily of at most $\\ell$ copies can be covered by a forest of copies—a set of copies of $F$ glued along single vertices or shared edges. For the negative results, the paper invokes an ordered-linear-hypergraph variant (Theorem 3.1) and a corollary (Corollary 3.4) that, for $(e,v)$-inseparable ordered hypergraphs, makes all induced copies respect the prescribed ordering. The positive direction relies on a set-mapping partition theorem to colour edges so that a monochromatic copy would contradict a maximal glued copy $(m,F)$, forcing the presence of the entire forest. The graphs $F$ of Theorem 1.4 arise from ordered linear 3-uniform hypergraphs $S$ by putting $xz$ in $E(F)$ whenever some $xyz \\in E(S)$ with $x < y < z$, which transfers forest structure from $S$ to $F$. The 2-density $m_2(F)$ controls Theorems 1.5 and 1.6: forests of copies do not raise $m_2$, while overlapping or cyclic unions of copies do, so Ramsey graphs must contain configurations heavier than $m_2(F)$.","core_discovery":"The central assertion is a dichotomy about unavoidable substructures in Ramsey graphs. Theorem 1.3 asserts that when $F$ is a cycle or a balanced complete multipartite graph with at least two edges, every forest of copies of $F$ is forced: for every such forest $\\mathcal{F}$ there is an $r \\ge 2$ such that every graph $G$ with $G \\xrightarrow{\\mathrm{n.n.i.}} (F)_r$ contains a subgraph isomorphic to $\\bigcup \\mathcal{F}$. Theorem 1.4 asserts that the dichotomy has a negative side: there exist infinitely many graphs $F$ such that for every $r \\ge 2$ there is a Ramsey graph $G \\to (F)_r$ in which all copies of $F$ are induced, and any two induced copies sharing an edge also share a third vertex not adjacent to either endpoint, so no non-trivial forest of copies occurs. Theorems 1.5 and 1.6 add that Ramsey graphs can be made locally non-Ramsey on every fixed vertex set and can contain $F$-free subgraphs with a positive fraction of the edges inside every subgraph, with Theorem 1.8 (every graph containing a cycle is Ramsey infinite) following from Theorem 1.5.","pith_inferences":["The dichotomy suggests a natural research direction: characterise the graphs $F$ for which all forests of copies are unavoidable; cycles and balanced complete multipartite graphs may be the first members of a larger family, and strict 2-balancedness or edge-transitivity could be the distinguishing invariant.","Because Theorems 1.4 and 1.6 rely on the unproved ordered-hypergraph variant (Theorem 3.1), supplying a proof of that variant would immediately secure the negative results; conversely, any failure of Theorem 3.1 would likely produce a concrete counterexample to Theorem 1.4.","The construction of $F$ from sum-constrained 3-uniform hypergraphs is tied to additive combinatorics; replacing the equations $x+y+z=n$ or $2n$ by other linear forms may yield families of graphs $F$ with even stronger local restrictions, such as forbidding any two copies from sharing a vertex."],"forward_implications":["For cycles and balanced complete multipartite graphs, every glued forest of copies is unavoidable in every Ramsey graph with sufficiently many colours.","For infinitely many dense, 4-connected graphs $F$, there are Ramsey graphs in which all copies are induced and edge-sharing copies always share a non-adjacent third vertex, so no non-trivial forest of copies exists.","Ramsey graphs for any graph containing a cycle can be made locally non-Ramsey: every $n$-vertex subgraph fails the two-colour non-induced Ramsey property for $F$, and consequently every such $F$ is Ramsey infinite.","For certain bipartite graphs $F$, every subgraph of a suitable Ramsey graph contains an $F$-free subgraph carrying at least $(\\ell(F)-1)/(2\\ell(F))$ of its edges."],"supporting_citations":[{"why":"Supplies the girth Ramsey theorem, the structural engine behind Theorems 1.5, 1.8, and the ordered hypergraph variant.","marker":"[10]"},{"why":"Provides the set-mapping partition theorem used to force forests of copies in Theorem 1.3.","marker":"[1]"},{"why":"Contributes the 2-density lower bound for Ramsey graphs that grounds Corollary 5.5 and Theorem 1.5.","marker":"[11]"}],"fun_headline_variants":["Ramsey dichotomy: cycles force forests, others avoid them","Cycles force forest subgraphs; some F allow none","Ramsey graphs: for cycles, all forests forced; for others, none","Some Ramsey graphs have no forest subgraphs; cycles force them","Dichotomy: cycles force every forest, some graphs avoid all"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the girth Ramsey theorem extends to ordered linear uniform hypergraphs (Theorem 3.1), a statement the paper invokes without proof; if that variant fails, the constructions behind the negative results lose their support.","fun_headline_variants_meta":{"raw":{"variants":["Ramsey dichotomy: cycles force forests, others avoid them","Cycles force forest subgraphs; some F allow none","Ramsey graphs: for cycles, all forests forced; for others, none","Some Ramsey graphs have no forest subgraphs; cycles force them","Dichotomy: cycles force every forest, some graphs avoid all"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000731,"raw_usage":{"total_tokens":3229,"prompt_tokens":863,"completion_tokens":2366,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":479,"completion_tokens_details":{"reasoning_tokens":2278}},"tokens_in":479,"tokens_out":2366,"duration_ms":17842,"temperature":1.0,"reasoning_tokens":2278,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T20:19:45.511014+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Two checks would settle the claims: (1) inspect whether the proof of the girth Ramsey theorem for graphs in [10] extends verbatim to ordered linear hypergraphs as Theorem 3.1 requires; (2) for $n=16$ and $r=2$, build the graph $F$ from Proposition 4.1 and the Ramsey graph $G$ from Corollary 3.4, then look for two induced copies of $F$ in $G$ that share an edge without a common third vertex non-adjacent to that edge, or for any non-induced copy of $F$—either would refute Theorem 1.4.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the set-mapping partition theorem used to force forests of copies in Theorem 1.3."},{"cited_title":"Rödl and A","cited_arxiv_id":null,"evidence_quote":"Contributes the 2-density lower bound for Ramsey graphs that grounds Corollary 5.5 and Theorem 1.5."}],"review_version":1}