{"id":"1f65f774-7ec8-40cd-a8d2-c363e42c3a3b","arxiv_id":"2607.26049","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":3.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Sublinear expansion—weak neighbourhood growth in sparse graphs—has resolved many long-standing extremal graph theory conjectures, and this survey organizes that progress.","lead":"This survey reviews a decade of progress in extremal graph theory driven by 'sublinear expansion,' a weak form of graph expansion that still gives strong connectivity. It shows how the technique has settled long-standing conjectures about subdivisions, cycle lengths, graph decompositions, and Latin squares.","discovery_kind":"review","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Survey's capstone 'resolutions' rest on two unpublished/forthcoming proofs; if either is flawed or not available, the central narrative overclaims.","rationale":"The reader's weakest assumption already identified the reliability of very recent or unpublished cited results, especially Montgomery's Ryser-Brualdi-Stein proof and the forthcoming harmonic-sum work. I agree this is the central soft spot. The survey otherwise functions well as an overview: no new derivations are attempted, and the historical attribution of results to sublinear expansion is consistent with the cited literature. However, the survey states these two capstone results as established facts without marking them as unverified preprints or forthcoming work. Because these results are used to justify the abstract's claim of 'resolution of many long-standing and notable problems', a flaw in either would directly undermine the central thesis. This does not warrant rejection, but it does warrant a conditional acceptance: the author should either confirm the correctness and availability of these proofs, or clearly label them as announced/preprint results rather than completed resolutions. This is a modest, non-adversarial revision that preserves the survey's value while preventing the narrative from overstating the current state of proof.","tokens_in":21710,"tokens_out":3916,"duration_ms":40087,"concrete_test":"Obtain arXiv:2310.19779 and the forthcoming harmonic-sum paper of Milojević, Montgomery, Pokrovskiy and Sudakov. If the harmonic-sum paper is not publicly available, the sentence in Section 5 cannot be verified and should be relabeled as an announced result. For [92], have an independent researcher verify the key auxiliary-graph lemma that decomposes the relevant graph into sublinear expanders and confirm that it implies the n-1 partial transversal theorem. If the lemma fails or the paper is unavailable, the survey's claim that Ryser-Brualdi-Stein is resolved for large even n should be softened to 'reported in a preprint'.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim is historical: sublinear expansion has led to the resolution of many long-standing problems. For most cited results this is well-supported, but two of the most prominent capstones are not established in the public literature. In Section 10, the survey states that Montgomery [92] showed every sufficiently large even-n Latin square has a partial transversal of size n-1. This is a preprint (arXiv:2310.19779) with a long proof, and the survey gives no independent verification. In Section 5, the survey reports that Erdős's harmonic-sum conjecture 'is true when d is sufficiently large' in forthcoming work of Milojević, Montgomery, Pokrovskiy and Sudakov, with no preprint, proof sketch, or reference to check. These two items are not peripheral: they are held up as recent successes of sublinear expansion and directly support the abstract's claim that the area has produced resolutions. If either proof has a serious gap or misstatement, the survey's central narrative is misleading. The concern is not that preprints cannot be cited, but that the survey presents these as established 'resolutions' without flagging their unreviewed or forthcoming status. This is the weakest load-bearing premise for the survey's thesis.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper is a survey of recent progress in extremal graph theory obtained through sublinear expansion. It introduces the relevant definitions (α-expanders, (ε,k)-expanders), describes the Komlós–Szemerédi framework, and then surveys applications to clique subdivisions and minors, cycle lengths, cycle decomposition and packing, nested and chorded cycles, rainbow Turán problems, Ramsey numbers of cycles, and two applications involving auxiliary graphs (cycles with all diagonals and Latin-square transversals). It closes with several open problems, including Mader's constant for subdivisions, Erdős–Gallai cycle decomposition, nested cycles without geometric crossings, rainbow-cycle bounds, and Hamiltonicity of regular sublinear expanders.","tokens_in":21973,"tokens_out":10907,"duration_ms":98123,"significance":"If the surveyed results are correct, the paper provides a valuable and readable synthesis of a technique that has become central to sparse extremal graph theory in the last decade. The survey is careful in its attributions, correctly separates published results from open problems, and repeatedly points to the companion technical survey of Letzter [78] for details. It also explicitly highlights several important open questions, which is a useful service. The author's own work appears prominently, but the presentation is generally measured. The main weakness is that two of the headline 'resolutions' advertised in the abstract rest on a preprint and a forthcoming paper, respectively; these need clearer epistemic status flags. With that local revision, the survey would be a reliable entry point to the field.","major_comments":[{"comment":"The sentence 'That this is true when d is sufficiently large is shown in forthcoming work of Milojević, Montgomery, Pokrovskiy and Sudakov' is presented as an established resolution of a long-standing conjecture, yet no reference, preprint identifier, or proof sketch is given. This result is explicitly used in the abstract's narrative of 'resolution of many long-standing and notable problems'. Please either supply a citable source (arXiv ID or accepted paper) or explicitly mark it as a recent announcement whose details are not yet public.","section":"Section 5 (Erdős harmonic-sum conjecture)"},{"comment":"The text states that 'Montgomery [92] showed that, when n is sufficiently large, every Latin square of order n contains a partial transversal with n−1 cells.' Although [92] is given an arXiv identifier, the surrounding wording ('showed', 'lengthy proof') presents it as a settled result. Since [92] is an unreviewed preprint (arXiv:2310.19779), the survey should qualify it as a preprint/announced result, e.g., 'announced in a preprint [92]' or 'proved in a recent preprint [92]', so that the reader can distinguish it from the refereed literature.","section":"Section 10 (Ryser–Brualdi–Stein conjecture)"}],"minor_comments":[{"comment":"The phrase 'showing d(t)=t^2+o(1)' appears to be a typo. The known bounds are d(t)=Θ(t^2), with lower bound (9/64+o(1))t^2 and upper bound (10/23+o(1))t^2 discussed later in the same section; an asymptotic equality with t^2 contradicts the lower bound. The intended statement is likely 'd(t)=O(t^2)' or 'd(t) ≤ (1+o(1))t^2'.","section":"Section 2 (paragraph before Theorem 2.1)"},{"comment":"The 'forthcoming work' of Milojević–Montgomery–Pokrovskiy–Sudakov is not listed in the references. If no preprint is yet available, at least add a reference entry with '(in preparation)' and the expected authors, so that the citation format is consistent with the rest of the survey.","section":"Section 5 (Erdős harmonic-sum conjecture)"},{"comment":"Consider adding a sentence in the introduction or in Section 10 clarifying that the survey reviews both published and unpublished (preprint/announced) results, and that the latter should be read with appropriate caution. This would preempt the ambiguity highlighted for the two capstone results.","section":"Section 10 (Ryser–Brualdi–Stein)"}],"recommendation":"minor_revision","confidential_remarks":"The stress-test concern raised by the reader is legitimate but, on reading the manuscript, it is a fixable transparency issue rather than a fatal flaw: the survey does disclose 'forthcoming' and gives an arXiv number for the preprint, but the wording is nevertheless too assertive. The high proportion of self-citations is natural given the author's central role, though the two unverified capstones make the acceptance decision partly dependent on the author's unpublished work. The journal may wish to confirm with the author the status of the Milojević–Montgomery–Pokrovskiy–Sudakov paper before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The thing to know: this is a genuinely useful survey of how sublinear expansion has become a central tool in extremal graph theory over the last decade. It is not a research paper—almost everything is attributed to prior work—but it earns its place as a roadmap. The author is a leader in the area, and the survey reflects that.\n\nThe best parts are the organization and the connections it draws. The definition of sublinear expansion and the historical arc from Komlós–Szemerédi through Liu–Montgomery's odd cycle result to the recent cycle packing and rainbow Turán work are presented clearly, with a good sense of which technical difficulties matter. The survey also flags several open problems (e.g., Conjecture 5.2, Question 5.1) and points readers to Letzter's technical survey for the details.\n\nNow the soft spots, in proportion. The reader's report and the stress-test both focus on the same issue: the two most prominent capstones—the large-n Ryser–Brualdi–Stein result and the harmonic-sum conjecture—rest on very recent or unavailable proofs. Section 10 cites a 2023 preprint for Ryser–Brualdi–Stein; Section 5 says the harmonic-sum result is 'forthcoming' with no preprint, no proof sketch, and no reference to check. The survey states both as established resolutions without flagging their status. That is a real weakness, since the abstract's claim that sublinear expansion has led to 'resolutions of many long-standing problems' leans on them. A careful reader should treat those two items as provisional. That does not sink the survey—the rest of the cited literature is solid, and the survey itself is careful elsewhere (e.g., it attributes the o(1) constants properly)—but it deserves a caveat.\n\nThe other concern is the heavy weight of the author's own papers in the highlighted results. For a survey by a leading contributor, that is expected, and the citations themselves look accurate. It is a minor bias risk, not a flaw.\n\nWho is this for? A graduate student or researcher wanting a map of recent progress in extremal graph theory, or someone looking for open problems. It deserves a serious referee: the survey is accurate, honest, and useful, and the only substantive fix is to hedge or annotate the two unpublished items. I would accept it after minor revisions.\n\nRecommendation: engage with it—cite it, assign it to a student, and send it to peer review with a request that the author mark the unpublished/forthcoming results as such.","headline":"A genuinely useful survey of sublinear expansion's recent impact, but two headline 'resolutions' still rest on unpublished or forthcoming work; worth refereeing with a request for caveats.","tokens_in":22425,"tokens_out":2539,"would_cite":true,"duration_ms":25293,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C35","05C38","05C48","05C75"],"pacs":[],"model":"deepseek-v4-flash","headline":"This survey argues that a weak form of expansion, in which neighbourhoods grow only logarithmically slowly, has become a central tool in extremal graph theory and lies behind a decade of breakthroughs from exact cycle lengths to Latin-squar","keywords":["sublinear expansion","expander graphs","extremal graph theory","graph subdivisions","graph minors","cycle lengths","cycle decomposition","Latin square transversals"],"falsifier":"Go to the proofs cited in Sections 5 and 10: the harmonic-sum cycle-length result announced as forthcoming, and the large-even Latin-square transversal proof. If either proof contains a gap that cannot be repaired—or if a counterexample appears, such as a sufficiently large even-order Latin square with no partial transversal of size n−1—the survey's central claim that sublinear expansion has resolved these problems would be false.","tokens_in":21589,"feed_emoji":"🕸️","tokens_out":11152,"duration_ms":105278,"temperature":0.7,"pith_summary":"Sublinear expansion is a deliberately weak connectivity property: in an n-vertex graph, every set of vertices of size up to n/2 has a neighbourhood at least proportional to |U|/log^2(3|U|/k). The survey's thesis is that this weak property, not the strong expansion of classical expander theory, has been the effective engine behind a wide range of recent results in extremal graph theory. The enabling fact is that any graph with large average degree contains a subgraph that is a sublinear expander and still has almost the same average degree, so a sparse graph can be replaced by an expander without losing the density that matters. Around this 'pass to an expander' step, the survey organises proofs of old conjectures about clique subdivisions, complete minors, cycle lengths and the odd cycle problem, cycle decompositions down to O(n log* n), rainbow cycles, Ramsey numbers, and even the existence of almost-full transversals in Latin squares. A sympathetic reader comes away with a single picture: weak expansion is a broadly applicable skeleton for constructing structure in sparse graphs.","feed_headline":"Sublinear expansion drives a decade of graph proofs","feed_subtitle":"One weak connectivity property, inherited by every dense enough graph, underlies results from exact cycle lengths to Latin-square transversa","key_machinery":"The central object is the (ε,k)-expander, a sublinear expander in which every vertex set U with k ≤ |U| ≤ n/2 has neighbourhood size at least (ε/log^2(3|U|/k))|U|. Its partner is a mid-1990s theorem: every graph of average degree d contains such a subgraph with average degree within (1−δ)d. The mechanism does the work by converting an arbitrary sparse graph into a graph whose iterated neighbourhoods grow, however slowly, in a controlled way; this gives paths between arbitrary vertices in poly-logarithmic length and, after careful construction, paths and cycles of exact or near-exact prescribed lengths.","core_discovery":"The paper's central claim is that sublinear expansion should be regarded as a unifying technique: many hard extremal problems become tractable once the graph is replaced by a sublinear expander. The load-bearing result is a theorem from the mid-1990s: every graph of average degree d contains a subgraph that is an (ε,k)-expander with average degree at least (1−δ)d, so the reduction costs almost nothing in density. The survey compiles the high-water marks of this approach: tight quadratic bounds for topological cliques, logarithmic-size complete minors in dense graphs, the resolution of the C4-free subdivision conjecture, the odd cycle problem and the sharp (1/2−o(1)) log n harmonic sum over c","pith_inferences":["If the pass-to-an-expander thesis generalizes as the survey suggests, hypergraph or directed-graph analogues of these problems may become tractable with a suitable notion of sublinear expansion; the survey does not make this claim.","The O(n log* n) barrier for cycle decomposition is implicitly presented as an artefact of iterative methods; a testable prediction is that a one-shot decomposition, if it exists, will need a structural characterisation of graphs that resist few-cycle decompositions.","The use of sublinear expansion in auxiliary graphs, for cycles with all diagonals and for Latin squares, suggests the technique may be most powerful when the problem's real difficulty has been hidden in a derived graph, and similar transfers could illuminate other Turán or colouring problems.","If random regular sublinear expanders turn out to be Hamiltonian, the survey's conjecture that every regular sublinear expander is Hamiltonian would place weak expanders on the same footing as the spectral expanders for which Hamiltonicity was recently proved."],"forward_implications":["Any graph with sufficiently large average degree contains a sublinear expander of almost the same degree, so the expander-replacement step is applicable to a broad class of extremal problems, not only those surveyed.","The cycle-length machinery implies that graphs of large chromatic number have odd cycles whose reciprocal lengths sum to at least (1/2−o(1)) log χ(G), which is optimal up to the o(1).","The cycle-decomposition results imply every n-vertex graph can be decomposed into O(n log* n) cycles and edges, and that improving this to O(n) likely requires a non-iterative or non-memoryless argument.","The Latin-square result implies every sufficiently large even-order Latin square has a partial transversal of size n−1, one short of a full transversal.","Conjectures stated in the survey, if true, would extend the same structural picture: every regular sublinear expander would be Hamiltonian, and every properly edge-coloured graph with no rainbow cycle would have O(n log n) edges."],"fun_headline_variants":["One expansion trick that settled many open graph questions","How a weak connectivity property cracked hard graph problems","Sublinear expansion: the key to a decade of graph breakthroughs","A survey of sublinear expansion's greatest hits in extremal graph theory","Sublinear expanders: the tool behind recent extremal graph wins"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The survey's narrative depends on the correctness of several very recent and still-unpublished results it cites—especially the proof of the n−1 partial transversal conjecture for all sufficiently large even-order Latin squares and the forthcoming resolution of the harmonic-sum cycle-length conjecture; if either is flawed, the survey's flagship examples of sublinear expansion at work would be unwarranted.","fun_headline_variants_meta":{"raw":{"variants":["One expansion trick that settled many open graph questions","How a weak connectivity property cracked hard graph problems","Sublinear expansion: the key to a decade of graph breakthroughs","A survey of sublinear expansion's greatest hits in extremal graph theory","Sublinear expanders: the tool behind recent extremal graph wins"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000721,"raw_usage":{"total_tokens":2995,"prompt_tokens":587,"completion_tokens":2408,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":331,"completion_tokens_details":{"reasoning_tokens":2325}},"tokens_in":331,"tokens_out":2408,"duration_ms":16513,"temperature":1.0,"reasoning_tokens":2325,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T00:43:31.673473+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Go to the proofs cited in Sections 5 and 10: the harmonic-sum cycle-length result announced as forthcoming, and the large-even Latin-square transversal proof. If either proof contains a gap that cannot be repaired—or if a counterexample appears, such as a sufficiently large even-order Latin square with no partial transversal of size n−1—the survey's central claim that sublinear expansion has resolved these problems would be false.","supporting_citations":[],"review_version":1}