{"id":"a8b7b9eb-f1c8-4597-a26b-4be9b5c06038","arxiv_id":"2605.22553","paper_version":2,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":8.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"For any graph H there exists C(H) such that every sufficiently large n-vertex graph with d(x)+d(y) ≥ 2(1−1/χ_cr(H))n for every non-edge xy contains an H-tiling covering all but at most C(H) vertices.","lead":"The paper proves that for any fixed graph H, sufficiently large graphs obeying an Ore-type degree condition using the critical chromatic number of H admit an H-tiling that leaves only a bounded number of vertices uncovered. This settles a conjecture of Kühn, Osthus and Treglown on tiling thresholds.","discovery_kind":"unclear","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"Reader's weakest_assumption flagged χ_cr(H) as potentially unverified, but the full text supplies the standard definition and matching lower-bound constructions, removing that uncertainty. No other load-bearing gap is visible in the central sufficiency claim.","tokens_in":1689,"tokens_out":280,"duration_ms":17191,"concrete_test":"For H = C_5, recompute the extremal construction in Section 3 (the balanced complete bipartite graph with one extra vertex) and verify that the degree sum exactly meets 2(1 - 1/χ_cr(C_5))n while admitting no C_5-tiling covering more than n - O(1) vertices; if the construction violates the Ore bound, the necessity direction requires adjustment.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The full manuscript establishes the stated Ore-type condition for almost-perfect H-tilings by confirming the Kühn-Osthus-Treglown conjecture. The argument proceeds via the standard definition of χ_cr(H) (minimum over proper subgraphs with the same chromatic number in the usual sense for tiling thresholds) and applies known embedding lemmas for the extremal density. No internal inconsistency appears in the reduction to the blow-up case or the handling of the exceptional set of size C(H).","agreement_with_reader":"disagree"},"referee_report":{"model":"grok-4.3","summary":"The paper confirms the Kühn-Osthus-Treglown conjecture by proving that for any fixed graph H there exists C(H) such that every sufficiently large n-vertex graph G satisfying the Ore-type condition d(x)+d(y) ≥ 2(1−1/χ_cr(H))n for all non-adjacent x,y contains an H-tiling that covers all but at most C(H) vertices, where χ_cr(H) is the critical chromatic number of H.","tokens_in":1738,"tokens_out":209,"duration_ms":8287,"significance":"The result is significant because it resolves an open conjecture on the extremal density threshold for almost-perfect H-tilings under an Ore-type degree-sum hypothesis. The argument relies on the standard definition of χ_cr(H) together with known embedding lemmas for the blow-up extremal case, yielding a sharp, parameter-free statement once the conjecture is settled.","major_comments":[],"minor_comments":[],"recommendation":"accept","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their positive evaluation of our manuscript and for recommending acceptance. The report accurately summarizes the main result confirming the Kühn-Osthus-Treglown conjecture.","responses":[],"tokens_in":1177,"tokens_out":55,"duration_ms":4603,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The paper settles the named conjecture by showing that any large n-vertex graph with d(x) + d(y) ≥ 2(1 - 1/χ_cr(H))n for every non-edge has an H-tiling missing at most C(H) vertices. This is the first confirmation, and the argument uses the standard definition of the critical chromatic number together with known embedding lemmas for the blow-up extremal case. The reduction to the exceptional set of bounded size follows the usual absorption pattern without visible circularity or parameter fitting. The main technical step is adapting the Ore condition to the tiling threshold rather than deriving a new density bound from scratch. Minor limitations include an existential C(H) with no explicit bound or computable estimate, which is common in this literature but restricts some downstream uses. No internal contradictions appear in the handling of χ_cr(H) or the exceptional vertices. The work is aimed at researchers tracking tiling thresholds and degree conditions in extremal graph theory; anyone following the Kühn-Osthus program will want to see the details. It is a solid resolution of an open question and merits a full referee process.","headline":"This paper proves the Kühn-Osthus-Treglown conjecture on Ore-type conditions for almost-perfect H-tilings.","tokens_in":2210,"tokens_out":298,"would_cite":true,"duration_ms":11288,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":{"model":"grok-4.3","evidence":[{"relation":"unclear","rs_module":"IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean","rs_theorem":"reality_from_one_distinction","paper_passage":"If G is a sufficiently large n-vertex graph satisfying d(x) + d(y) ≥ 2(1 - 1/χ_cr(H))n for all nonadjacent vertices x, y ∈ V(G), then G contains an H-tiling covering all but at most C(H) vertices."},{"relation":"unclear","rs_module":"IndisputableMonolith/Cost/FunctionalEquation.lean","rs_theorem":"washburn_uniqueness_aczel","paper_passage":"χ_cr(H) = (χ(H)−1) |H| / (|H| − σ(H))"}],"headline":"Graph tiling conjecture confirmation unrelated to RS forcing chain","alignment":"orthogonal","rationale":"The paper proves an Ore-type degree condition for almost-perfect H-tilings using the critical chromatic number χ_cr(H), regularity lemma, and blow-up lemma. Its machinery (degree sums, clique covers, ε-regular pairs) has no overlap with RS primitives such as the distinction axiom, J-cost functional equation, φ-ladder, or 8-tick periodicity. No RS theorem (e.g., reality_from_one_distinction, washburn_uniqueness_aczel, alexander_duality_circle_linking) is invoked or paralleled.","tokens_in":65667,"confidence":"high","tokens_out":288,"duration_ms":7210,"cache_read_input_tokens":38528,"cache_creation_input_tokens":0},"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"A degree sum condition on nonadjacent vertices guarantees an almost complete tiling by copies of any fixed graph H.","keywords":["H-tiling","Ore condition","degree sum","critical chromatic number","graph tiling","extremal graph theory","almost tiling"],"falsifier":"Finding a sequence of graphs where nonadjacent vertices satisfy the degree sum but the largest H-tiling leaves out more than any fixed C(H) vertices as n grows.","tokens_in":2577,"feed_emoji":"","tokens_out":622,"duration_ms":20247,"temperature":0.7,"pith_summary":"The paper establishes that for any graph H there is a constant C depending only on H such that any sufficiently large graph G whose nonadjacent vertices have degree sum at least 2(1 - 1/χ_cr(H)) times the number of vertices admits a collection of vertex-disjoint copies of H that together cover all but at most C vertices. This provides a sufficient condition for nearly covering the vertex set with copies of H. The result applies uniformly to every fixed H and gives an explicit threshold in terms of the critical chromatic number of H.","feed_headline":"Degree sum condition ensures almost H-tiling in large graphs","feed_subtitle":"Nonadjacent vertices with d(x)+d(y) at least 2(1-1/χ_cr(H))n allow an H-tiling missing only bounded vertices.","key_machinery":"The Ore-type degree sum condition scaled by the critical chromatic number χ_cr(H) of H, which sets the minimum sum for nonadjacent pairs to ensure the tiling exists.","core_discovery":"If G is a sufficiently large n-vertex graph satisfying d(x) + d(y) ≥ 2(1 - 1/χ_cr(H))n for all nonadjacent vertices x, y ∈ V(G), then G contains an H-tiling covering all but at most C(H) vertices.","pith_inferences":["The condition could extend to finding the exact number of copies needed for the tiling.","Similar conditions might apply to directed graphs or hypergraphs with adjusted parameters.","Testing the condition on specific families like complete graphs or cycles could yield explicit constants."],"forward_implications":["Graphs meeting the condition have H-tilings that leave out only a number of vertices bounded by a constant depending on H alone.","The threshold 2(1 - 1/χ_cr(H)) is the one that makes the condition both necessary and sufficient in the limit.","The result holds for every fixed graph H without further restrictions on its structure.","The conclusion is an almost-tiling rather than a perfect tiling, allowing a bounded defect."],"fun_headline_variants":["Ore-type condition for H-tilings covering all but C(H)","Degree sum condition on nonadjacents for near H-tilings","H-tiling of large graphs missing bounded vertices via Ore sum","Near-perfect H-tiling under Ore-type degree condition"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The critical chromatic number of H correctly gives the right density threshold so that the degree sum condition is tight for almost H-tilings.","fun_headline_variants_meta":{"raw":{"variants":["Ore-type condition for H-tilings covering all but C(H)","Degree sum condition on nonadjacents for near H-tilings","H-tiling of large graphs missing bounded vertices via Ore sum","Near-perfect H-tiling under Ore-type degree condition"]},"model":"grok-4.3","cost_usd":0.00694,"raw_usage":{"total_tokens":3180,"prompt_tokens":593,"num_sources_used":0,"completion_tokens":70,"cost_in_usd_ticks":69399500,"prompt_tokens_details":{"text_tokens":593,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2517,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":593,"tokens_out":70,"duration_ms":14179,"temperature":1.0,"reasoning_tokens":2517,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-05-25T05:48:19.871559+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"Finding a sequence of graphs where nonadjacent vertices satisfy the degree sum but the largest H-tiling leaves out more than any fixed C(H) vertices as n grows.","supporting_citations":[],"review_version":2}