{"id":"b2299e05-cb4a-461e-afcb-65736b8e4c97","arxiv_id":"2607.16854","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For several r-uniform linear hypertrees with four edges, the maximum number of edges in a linear r-uniform hypergraph avoiding them is determined; the 4-uniform 4-edge path case is settled exactly.","lead":"This paper determines exact edge-number limits for linear hypergraphs that avoid certain small tree-like configurations, extending earlier work on 3-uniform hypergraphs to every uniformity r. The headline result settles the 4-uniform, 4-edge path case and characterizes the extremal examples as Steiner systems, while also correcting a flawed earlier proof.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 2's lower-bound construction requires an affine plane of order r-1, which does not exist for r=7; the theorem as stated is unproven for such r.","rationale":"The central Theorem 7 proof was checked in detail: the minimal-counterexample reduction, the trace lemmas, the case analysis for Δ=6, and the cograph/edge-decomposition argument for equality all appear internally sound. The B_4^r and E_4^r upper bounds also check out. The one genuine flaw is Proposition 2's unsupported existence assumption: its construction requires an affine plane of order r-1, which does not exist for all r satisfying the stated divisibility condition. The reader's weakest_assumption identifies exactly this issue. Since the reader's CONDITIONAL verdict already reflects this concern, no verdict change is needed.","tokens_in":20419,"tokens_out":44593,"duration_ms":369199,"concrete_test":"Specialize to r=7, n=43 (so (r-1)^2 = 36 divides n-r = 36). Check whether any linear 7-uniform E_4^7-free hypergraph on 43 vertices with 42 edges exists, or whether the proof can be carried out without an affine plane of order 6. Since BRC excludes such a plane, and an affine plane is equivalent to a net of degree q+1=7, no such A exists; if no alternative construction is supplied, Proposition 2 remains unproven as stated and should be revised to require q=r-1 to be a prime power.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Section 5, Proposition 2 claims ex_lin_r(n,E_4^r) ≥ r(n-r)/(r-1) under only the divisibility hypothesis (r-1)^2 | (n-r). Its proof begins 'Let A be an affine plane of order q' with q=r-1, and uses its q+1=r parallel classes. An affine plane of order q exists iff a projective plane of order q exists, and the Bruck–Ryser–Chowla theorem rules out q=6. Hence for r=7 (and any r with r-1 not a prime power), the required A does not exist, so the construction is undefined. The hypothesis (r-1)^2 | (n-r) does not imply such an A exists. Thus Proposition 2 is not established for those r. This is load-bearing for the paper's crown lower-bound claim, though it does not affect the r=4 central P_4^4 theorem or the upper bounds in Theorems 3/4. The statement should be amended to assume r-1 is a prime power (or that S(2,r-1,(r-1)^2) exists).","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies linear Turán numbers of r-uniform linear hypertrees. It proves the exact linear Turán number of the uniform star S_k^r, gives a general lower-bound construction for arbitrary linear hypertrees conditional on the existence of certain Steiner systems, determines the exact extremal value and extremal structure for the broom B_4^r, obtains an upper bound and a lower-bound construction for the crown E_4^r, and provides a lower bound for P_4^r together with a sharp bound for connected P_4^r-free hypergraphs under degree hypotheses. For r=4, the paper exhibits counterexamples to a structural claim in the earlier proof of Zhang and Wang, proves ex_4^{lin}(n,P_4^4) ≤ 5n/4, and characterizes equality as disjoint unions of S(2,4,16).","tokens_in":20660,"tokens_out":21173,"duration_ms":183022,"significance":"The main contribution is the sharp 4-uniform result for P_4^4, with a full extremal characterization, and the exact linear Turán number for the broom B_4^r. The paper also identifies a concrete gap in a previously published proof, which is a useful service to the area. The proofs are elementary and, apart from the issue discussed below, the trace-based case analysis in Theorem 7 appears sound. The extremal characterization via line graphs and cographs is elegant. However, one lower-bound claim, Proposition 2, is overstated because it silently assumes the existence of an affine plane of order r-1; this affects the crown lower-bound claim for infinitely many r but not the central P_4^4 theorem.","major_comments":[{"comment":"Proposition 2 is stated for all r≥3 under only the divisibility condition (r−1)^2 | (n−r), but its proof begins 'Let A be an affine plane of order q' with q=r−1. An affine plane of order q exists only when S(2,q,q^2) does, and in particular only if q is a prime power or another admissible order. For r=7, q=6, the Bruck–Ryser–Chowla theorem rules out such a plane; the paper itself notes in §1.3 that no S(2,6,36) exists. Thus the construction is undefined for r=7 and for other non-prime-power values of r−1, and the claimed lower bound ex_lin_r(n,E_4^r) ≥ r(n−r)/(r−1) is not established under the stated hypotheses. This is load-bearing for the crown lower-bound claim, though it does not affect Theorem 4 or Theorem 7. I recommend restating Proposition 2 with the explicit existence assumption of an affine plane of order r−1, or equivalently S(2,r−1,(r−1)^2), and marking the lower bound as con","section":"§5, Proposition 2"}],"minor_comments":[{"comment":"In the displayed computation of (r+1)n/r − |E(H)|, the term |T|/r is written as |T|+2/r in the third line. The previous line has |T|/r correctly. The conclusion is unaffected, but the displayed algebra should be corrected.","section":"§7, proof of Theorem 6"},{"comment":"The sentence 'Since G is a connected cograph, G must be disconnected. Clearly G is m−17 regular' is confusing as written: it should refer to the complement of G, not to G itself. With 'complement' substituted, the component argument is correct.","section":"§8.2, extremal characterization in Theorem 7"},{"comment":"The equality statement reads 'equality holds if and only if the hypergraphs is disjoint union of Steiner systems'; this should be 'the hypergraph is a disjoint union'.","section":"§8, Theorem 7 statement"},{"comment":"The sentence 'From the above claim |V(H_i)| ≥ (r−1)k+1 = t+(r−1)' is misstated. The component has |V(H_i)|=t, while a copy of T_r^k would require (r−1)k+1 > t vertices. The comparison should be made between these two quantities, not written as a lower bound on |V(H_i)|.","section":"§3, proof of Theorem 2"}],"recommendation":"major_revision","confidential_remarks":"The core results, especially Theorem 7 and Theorem 3, appear sound and are likely publishable after revision. The Proposition 2 issue is real but local: the missing existence hypothesis is easy to add, and the central r=4 theorem is unaffected. I do not see grounds for rejection. The proof of Theorem 7 is long; I checked the main case structure and found no fatal gap, though the notation around complements in the cograph argument should be fixed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: this is a legitimate and mostly sound paper. The headline results are the sharp bound for P_4^4 with the extremal characterization, the broom bound with equality iff disjoint unions of S(2,r,r^2), and the general lower-bound template for arbitrary linear hypertrees. I checked the star bound, the Steiner-system constructions, the broom degree arguments, the crown counting proof, and the P_4^4 trace analysis; the mathematics holds up. The counterexamples to Zhang and Wang are explicit and correct.\n\nThe one real problem is Proposition 2. It promises ex_lin_r(n,E_4^r) ≥ r(n-r)/(r-1) under only the divisibility condition (r-1)^2 | (n-r), but the proof begins \"Let A be an affine plane of order q\" with q=r-1. An affine plane of order 6 does not exist (Bruck–Ryser–Chowla), so for r=7 the construction is undefined. The paper itself notes later that no S(2,6,36) exists, so this is an internal miss. The theorem as stated is unproven for r=7 and for any r where r-1 is not a prime power. The fix is easy—add the affine-plane existence assumption (or r-1 a prime power) to the hypothesis—but it needs to be made. This does not affect Theorem 4's upper bound or the P_4^4/broom results, and the crown lower bound is only a constant-factor gap anyway, so the damage is limited.\n\nMinor issues: complement-graph notation is confusing in a few places, there's an algebra display typo, and Theorem 6's statement says \"We proof.\" Trivial to fix.\n\nOn the citation pattern: nothing suspicious. The self-citation to the authors' earlier crown paper is technique-only and appropriate; the use of Keevash's design existence theorem is standard.\n\nWho is this for? Anyone working on linear Turán numbers or design-based extremal constructions. It deserves a serious referee; the main theorems are valuable and the flaws are repairable. I'd send it to review with a specific request to fix Proposition 2 and double-check the long case analysis in Theorem 7.","headline":"Worth refereeing: P_4^4 result is clean, but the crown lower bound in Proposition 2 quietly assumes an affine plane of order r-1, which fails for r=7.","tokens_in":21137,"tokens_out":2639,"would_cite":true,"duration_ms":23714,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C65","05C35","05B05"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that the maximum number of edges in a 4-uniform linear hypergraph avoiding the 4-edge path is 5n/4, attained only by disjoint unions of Steiner systems S(2,4,16), and corrects an earlier proof by exhibiting counterexamples","keywords":["linear Turán number","uniform linear hypergraph","hypertree","Steiner system","extremal hypergraph","4-uniform path","crown","broom"],"falsifier":"Find a connected 4-uniform linear hypergraph on 20 vertices with more than 25 hyperedges and no four hyperedges forming a P_4^4; the paper's proof says this is impossible, so its existence would falsify the bound. Equivalently, exhibit a connected 16-regular cograph on 25 vertices decomposable into 20 edge-disjoint K_5s, which the paper's counting rules out.","tokens_in":20278,"feed_emoji":"🌳","tokens_out":10715,"duration_ms":81597,"temperature":0.7,"pith_summary":"This paper is trying to settle the linear Turán number for several uniform hypertrees with four hyperedges: the broom, the crown, and the 4-edge path. Its sharpest result is for the 4-uniform path P_4^4, where it shows that any P_4^4-free linear 4-uniform hypergraph on n vertices has at most 5n/4 hyperedges, with equality exactly for disjoint unions of the Steiner system S(2,4,16). This is the 4-uniform case of a conjecture that would unify bounds for paths of low edge count, and it corrects a previous proof by giving explicit counterexamples to a key structural claim. Why care: exact answers in linear Turán theory are rare, and the extremal objects turn out to be classical designs, connecting extremal hypergraph theory to design theory.","feed_headline":"Uniform hypergraphs dodging the 4-edge path cap at 5n/4","feed_subtitle":"Extremal hypergraphs are exactly disjoint unions of Steiner systems S(2,4,16); the paper fixes a gap in an earlier proof.","key_machinery":"The load-bearing objects are Steiner systems S(2,r,n): linear r-uniform hypergraphs in which every pair of vertices lies in exactly one hyperedge. They saturate the pair-counting upper bound and provide all extremal constructions, including the lower bound for the path and the equality cases for the broom and the 4-uniform path. For the P_4^4 proof, the machinery is the line graph L(H) of the hypergraph: linearity and P_4^4-freeness make L(H) a connected cograph (a graph with no induced path on 4 vertices), 5-regularity makes it 16-regular, and an edge-decomposition of L(H) into n copies of K_5 — one per vertex of H — turns a counting contradiction on the cograph's connected components into","core_discovery":"The central claim is that the linear Turán number of the 4-uniform path P_4^4 is at most 5n/4, with equality exactly when the hypergraph is a disjoint union of Steiner systems S(2,4,16). More broadly, the paper determines the linear Turán number of the broom B_4^r as (r+1)n/r when S(2,r,r^2) exists, characterizes extremal hypergraphs as disjoint unions of that Steiner system, and gives upper and lower bounds for the crown E_4^r that leave only a constant-factor gap. For the path P_4^r it constructs dense examples from S(2,r,r^2) reaching (r+1)n/r and conjectures sharpness. For r=4, it exhibits counterexamples to a structural claim (V(S_k)=V(H)) in an earlier proof, then gives a new proof: a","pith_inferences":["One testable extension: if the line-graph/cograph argument for r=4 generalizes to other r where S(2,r,r^2) exists, the conjectured bound (r+1)n/r for P_4^r may reduce to a similar regularity plus cograph decomposition; the key check is whether the extremal hypergraph must be (r+1)-regular.","The gap in the crown lower bound — it needs an affine plane of order r-1, not just the stated divisibility condition — suggests the bound may fail for r=7 and motivates a search for alternative designs or a corrected hypothesis.","The paper's recurring theme, Steiner systems as the unique extremal objects, hints that for many linear hypertrees exact Turán results might follow from design-existence theorems, making the conditional results unconditional for all sufficiently large admissible n.","A direct project is to decide Conjecture 1 for r=5 or r=8, where S(2,r,r^2) exists, by attempting the same cograph decomposition and checking whether any non-design extremal hypergraphs appear."],"forward_implications":["If correct, ex_4^lin(n, P_4^4) = 5n/4 is fully resolved for all n admitting a partition into 16-vertex blocks, with a complete equality characterization.","The exact value ex_r^lin(n, B_4^r) = (r+1)n/r holds whenever S(2,r,r^2) exists, and disjoint unions of that Steiner system are the only extremal hypergraphs.","The crown E_4^r has (2r-1)n/r as an upper bound, and the new lower construction shows the true value lies within a constant factor, not exactly pinned.","For every linear r-uniform hypertree T_k^r, the construction shows the linear Turán number is at least n(k-1)/r whenever the divisibility and design-existence conditions are met.","The counterexamples show the earlier proof of the 4-uniform path bound is not salvageable as written, so the new argument is the justification for the bound."],"fun_headline_variants":["Exact linear Turan number for 4-uniform paths: 5n/4","Steiner systems nail exact bound for 4-uniform paths","New proof resolves 4-uniform path Turan bound","Exact extremal hypergraphs for 4-path: Steiner systems","Turan number for 4-uniform paths settled at 5n/4"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The crown lower bound is stated under only a divisibility condition, but its proof requires an affine plane of order r-1; such planes exist only when r-1 is a prime power, and for r=7 this fails, so the theorem as stated is not established for all r it claims to cover.","fun_headline_variants_meta":{"raw":{"variants":["Exact linear Turan number for 4-uniform paths: 5n/4","Steiner systems nail exact bound for 4-uniform paths","New proof resolves 4-uniform path Turan bound","Exact extremal hypergraphs for 4-path: Steiner systems","Turan number for 4-uniform paths settled at 5n/4"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000896,"raw_usage":{"total_tokens":3874,"prompt_tokens":1099,"completion_tokens":2775,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":843,"completion_tokens_details":{"reasoning_tokens":2693}},"tokens_in":843,"tokens_out":2775,"duration_ms":20604,"temperature":1.0,"reasoning_tokens":2693,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T19:54:35.455437+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find a connected 4-uniform linear hypergraph on 20 vertices with more than 25 hyperedges and no four hyperedges forming a P_4^4; the paper's proof says this is impossible, so its existence would falsify the bound. Equivalently, exhibit a connected 16-regular cograph on 25 vertices decomposable into 20 edge-disjoint K_5s, which the paper's counting rules out.","supporting_citations":[],"review_version":1}