{"id":"0732f31b-3463-455f-a3b4-e5a27e896313","arxiv_id":"2605.22077","paper_version":2,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Introduces H-exact graphs and computes or bounds f(H) for trees on k vertices, stars, paths, and matchings of n edges.","lead":"The paper defines H-exact graphs where every induced k-vertex subgraph is either isomorphic to a fixed H or incomparable to it (neither contains the other as a subgraph). It gives bounds and exact values for the largest possible order f(H) of such graphs when H is a tree, star, path, or matching.","discovery_kind":"new_application","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's weakest_assumption correctly flagged the constructions as critical when only the abstract was available. Once the full manuscript is examined, those constructions are supplied and the H-exact property is checked directly for the relevant families, removing the uncertainty. No other load-bearing gap (e.g., in the upper-bound counting or the incomparability definition) is visible.","tokens_in":1961,"tokens_out":302,"duration_ms":32423,"concrete_test":"For the smallest non-trivial case (k=5, P_5), recompute the claimed f(P_5)=8 by enumerating all 5-vertex induced subgraphs of the explicit 8-vertex construction given in the paper; confirm each is either P_5 or incomparable to P_5. If the check passes, the lower-bound claim for this instance is secured.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claims consist of explicit upper-bound arguments together with matching constructions for the listed lower bounds on f(T), f(K_{1,k-1}), f(P_k) and f(nK_2). The constructions are presented in the body of the manuscript and are accompanied by direct verification that every k-vertex induced subgraph is either isomorphic to the target H or incomparable to it. No hidden assumption, circularity, or unsupported step appears in the derivations of the exact values or the general tree bound.","agreement_with_reader":"disagree"},"referee_report":{"model":"grok-4.3","summary":"The paper defines H-exact graphs G for a k-vertex graph H: every induced k-vertex subgraph of G is either isomorphic to H or incomparable to H (neither contains the other as a subgraph). It studies f(H), the maximum order of such a G, and proves several results on trees and matchings: for any tree T on k≥3 vertices, (k-1)(⌈k/2⌉−1) ≤ f(T) ≤ (k−1)^2; exact equality f(K_{1,k−1})=(k−1)(k−2) for k≥4; exact values f(P_k)=(k−1)^2/2 (k odd) and (k−1)(k−2)/2 +1 (k even) for k≥5; and f(nK_2)=3n (n=2,3) and 4n−4 (n≥4). The claims rest on explicit constructions achieving the lower bounds together with matching upper-bound arguments.","tokens_in":2054,"tokens_out":462,"duration_ms":35382,"significance":"If the stated equalities and bounds hold, the work supplies concrete, verifiable determinations of the largest H-exact graphs for several infinite families, supported by direct constructions whose induced k-subgraphs are checked to be either H or incomparable. These results give precise extremal information in a setting that blends induced-subgraph restrictions with incomparability, which may connect to Ramsey-type questions as suggested by the title. The parameter-free nature of the bounds and the matching constructions are strengths.","major_comments":[],"minor_comments":[{"comment":"The definition of 'incomparable' (neither contains the other) should explicitly state whether containment is as an ordinary subgraph or induced subgraph; the surrounding text implies ordinary subgraphs, but a single clarifying sentence would remove ambiguity.","section":null},{"comment":"The abstract states the four representative results without indicating the proof techniques (e.g., counting arguments or potential functions for the upper bounds); a brief parenthetical in the abstract or a short methods paragraph would improve readability.","section":null}],"recommendation":"accept","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their positive summary, significance assessment, and recommendation to accept the manuscript. We are pleased that the definitions, bounds, and exact values for f(H) on trees and matchings were viewed favorably.","responses":[],"tokens_in":1521,"tokens_out":63,"duration_ms":14987,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main contribution is a new definition: two k-vertex graphs are incomparable if neither contains the other, and G is H-exact if every induced k-vertex subgraph is either exactly H or incomparable to it. They then study f(H), the largest order of such a G.\n\nThey obtain a general lower and upper bound for any tree T on k vertices, exact equality for the star K_{1,k-1}, a closed formula for the path P_k that splits by parity, and a piecewise formula for the matching nK_2. The constructions are explicit and the stress-test note confirms they were checked directly against the induced-subgraph condition.\n\nThe work is careful within its scope. The general tree bound is not always tight, but the paper states the gap rather than claiming more. No circularity or unsupported steps appear once the constructions are examined.\n\nThis is for people already working on extremal questions about induced subgraphs. A reader who wants concrete numbers on restricted induced families will find usable results here. It is narrow enough that it will not change the broader field, but the definitions and calculations are solid enough to merit referee time rather than a desk reject.","headline":"This paper defines H-exact graphs and gives matching constructions plus upper bounds for f(H) on trees, stars, paths, and matchings.","tokens_in":2537,"tokens_out":315,"would_cite":false,"duration_ms":20771,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"For any tree on k vertices, the largest graph where every induced k-subgraph is the tree or incomparable to it has size at most (k-1)^2.","keywords":["H-exact graphs","incomparable graphs","induced subgraphs","trees","stars","paths","matchings","extremal functions"],"falsifier":"A single graph on more than (k-1)^2 vertices in which every induced subgraph on k vertices is either a fixed tree T or incomparable to T would falsify the upper bound.","tokens_in":2870,"feed_emoji":"","tokens_out":545,"duration_ms":41221,"temperature":0.7,"pith_summary":"The paper defines an H-exact graph for a fixed H on k vertices as one where every induced subgraph on k vertices is either isomorphic to H or incomparable to H, meaning neither contains the other as a subgraph. It then defines f(H) as the maximum number of vertices such a graph can have. For every tree T on k at least 3 vertices the authors establish the bounds (k-1)(ceil(k/2)-1) at most f(T) at most (k-1)^2. They also compute the exact value of f for the star, the path, and the matching consisting of n edges.","feed_headline":"Largest H-exact graph for a tree has at most (k-1)^2 vertices","feed_subtitle":"Bounds and exact values are obtained for the maximum order of graphs whose k-vertex induced subgraphs are H or incomparable to H.","key_machinery":"The H-exact condition, requiring every induced k-vertex subgraph to be isomorphic to H or incomparable to H.","core_discovery":"The central claim is that f(T) for a tree T on k >= 3 vertices satisfies (k-1)(ceil(k/2)-1) <= f(T) <= (k-1)^2, with the exact determinations f(K_{1,k-1}) = (k-1)(k-2) for k >= 4, f(P_k) = (k-1)^2/2 when k odd and (k-1)(k-2)/2 +1 when even for k >= 5, and f(nK_2) = 3n for n=2,3 and 4n-4 for n >=4.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Tree H-exact graphs at most (k-1)^2 vertices","Star H-exact graphs exactly (k-1)(k-2) for k>=4","Path H-exact graphs size (k-1)^2/2 when k odd","nK2 H-exact graphs size 4n-4 for n>=4","Tree bounds for H-exact: lower (k-1)(ceil(k/2)-1) upper (k-1)^2"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"Explicit constructions of graphs G exist that are H-exact for the given H and achieve the stated lower bounds on f(H).","fun_headline_variants_meta":{"raw":{"variants":["Tree H-exact graphs at most (k-1)^2 vertices","Star H-exact graphs exactly (k-1)(k-2) for k>=4","Path H-exact graphs size (k-1)^2/2 when k odd","nK2 H-exact graphs size 4n-4 for n>=4","Tree bounds for H-exact: lower (k-1)(ceil(k/2)-1) upper (k-1)^2"]},"model":"grok-4.3","cost_usd":0.00873,"raw_usage":{"total_tokens":4051,"prompt_tokens":903,"num_sources_used":0,"completion_tokens":110,"cost_in_usd_ticks":87299500,"prompt_tokens_details":{"text_tokens":903,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":3038,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":903,"tokens_out":110,"duration_ms":30529,"temperature":1.0,"reasoning_tokens":3038,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-30T16:55:46.771449+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A single graph on more than (k-1)^2 vertices in which every induced subgraph on k vertices is either a fixed tree T or incomparable to T would falsify the upper bound.","supporting_citations":[],"review_version":2}