{"id":"adab5f06-11ef-4dab-b47f-a89e57a5e175","arxiv_id":"2607.28241","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For every graph T^s_{r,k} formed by k copies of K_r sharing a clique of order s, the homomorphism threshold equals (2r-5)/(2r-3).","lead":"The paper finds the exact homomorphism threshold for an infinite family of non-complete graphs built by gluing cliques along a shared core. This is the first such exact determination beyond ordinary cliques, tightening a classical density-to-structure question in extremal graph theory.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The reader's weakest-assumption note correctly flags the external quantitative clique-homomorphism theorem and residual risk on the long counting induction; both are real but already priced into the MODERATE / medium-risk ACCEPT. After checking the places where those dependencies are used (Proposition 3.3 only needs the homomorphism half of Theorem 2.5; Lemma 3.4's induction closes by reduction of overlap or direct production of T_{r,k}), no stronger load-bearing gap appears. The s≥ 2 case is short and clean; the r=3 appendix is independent and elementary. Consequently the verdict and confidence level require no adjustment.","tokens_in":19315,"tokens_out":547,"duration_ms":41005,"concrete_test":"Re-read the a=1 terminal case of Lemma 3.4 (final two pages) and verify that the successive applications of Lemma 2.2 inside G_c with the pre-chosen constants λ_j produce a heavy pair {p',q'} that cannot be {x_2,y_2} (by Claim 3.7) and cannot lie outside F, forcing {w_1,c} and the final replacement contradiction; if any constant inequality fails for large n the induction collapses.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim (Theorem 1.1) is supported by a complete case split: lower bound from the known chromatic-threshold classification, short reduction to the clique case when s≥ 2 (Claim 3.1 + Theorem 2.5), and a structured heavy-edge / partition / matching argument when s=1 that ultimately produces an explicit bounded T_{r,k}-free blow-up quotient. The longest internal step (induction on overlap a inside Lemma 3.4) is intricate but locally self-contained; each case either produces a forbidden T_{r,k} via Claim 3.6 or reduces the overlap while preserving the structural hypotheses needed for the inductive step. The external black-box (Theorem 2.5) is invoked only for its homomorphism-direction bound on K_r-free and K_{r-1}-free dense graphs, not for the maximality-to-blow-up clause, and the quantitative tower dependence is absorbed into the ε-dependent constant permitted by the definition of δ_hom. No internal inconsistency or missing case that would falsify the equality was located.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper determines the homomorphism threshold exactly for the infinite family T^s_{r,k} (k copies of K_r glued along a common K_s): δ_hom(T^s_{r,k}) = (2r−5)/(2r−3) for all r≥3, k≥1, 1≤s≤r. The lower bound is immediate from the known chromatic-threshold classification. For s≥2 the upper bound reduces to the clique case via a short clique-counting argument showing that dense T^s_{r,k}-free graphs are already K_r-free. For s=1 the authors prove a stronger blow-up statement for maximal dense T_{r,k}-free graphs, via a heavy-edge vertex cover of bounded size, a refined independent-set partition obtained from the clique homomorphism theorem, and a matching/König argument that produces a bounded T_{r,k}-free quotient; the longest step is an induction on blade overlap inside Lemma 3.4.","tokens_in":19518,"tokens_out":893,"duration_ms":32808,"significance":"Prior to this work the only graphs with exactly known homomorphism thresholds were cliques. The paper supplies the first infinite family of non-complete forbidden graphs for which δ_hom is determined exactly, and shows that the clique value is stable under gluing several K_r’s along a common clique. The common-neighborhood clique-counting lemmas (especially the inductive amplification Lemma 2.2) are of independent technical interest and cleanly extend the dense-edge phenomenon of Fox–Wigderson. The argument is fully combinatorial and self-contained once the external clique blow-up theorem is granted as a black box.","major_comments":[],"minor_comments":[{"comment":"The induction on the overlap parameter a inside Lemma 3.4 (pp. 14–16) is correct but dense; a short roadmap paragraph at the start of §3.2 listing the cases (a=0; a≥1 with core outside W; a≥2 with core in W; a=1 with core in W) would help the reader track the reductions.","section":"§3.2, Lemma 3.4"},{"comment":"The quantitative dependence of α_h and the λ_j sequence on ε is only sketched via successive applications of Lemmas 2.1–2.3. A one-line remark that all constants remain positive and depend only on r,ε (so that the tower-type bound of Theorem 2.5 is absorbed into the ε-dependent order permitted by δ_hom) would make the bookkeeping fully explicit.","section":"§3.1, Choice of the heavy-edge constant"},{"comment":"Theorem 2.5 is cited as an arXiv preprint. If a published version exists, update the reference; otherwise a brief parenthetical note that only the homomorphism-direction bound (not the maximality-to-blow-up clause) is used would clarify the logical dependence.","section":"§2, Theorem 2.5"},{"comment":"In the appendix (r=3) the constant 6k²/ε for |V(F)| is slightly looser than the main-text 4k²/α_h pattern; aligning the two presentations would improve uniformity.","section":"Appendix A"},{"comment":"Minor typos: “Erd˝ os” spacing inconsistencies; “K¨ onig” accent; “hom− − →” rendering; “therparts” (p. 12) and “Sincetis” (p. 6) missing spaces. A global proof-reading pass is recommended.","section":null}],"recommendation":"minor_revision","confidential_remarks":"The central argument is sound and the result is a genuine advance. The only external load-bearing ingredient is the recent clique blow-up theorem (arXiv:2403.17910); the manuscript uses it correctly and only in the homomorphism direction. I see no obstacle to acceptance after routine polishing."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The headline result is real: they pin δ_hom exactly at (2r−5)/(2r−3) for every T^s_{r,k} (k copies of K_r glued on a K_s). Before this, exact values were only known for cliques. That is the right next step after Goddard–Lyle and the odd-cycle contrast (positivity only).\n\nWhat works. Lower bound is free from the Allen–Böttcher et al. chromatic-threshold classification. For s≥2 the upper bound is short: dense T-free graphs are already K_r-free by the counting lemma, then invoke the known clique homomorphism theorem. The reusable piece is Lemma 2.2 (and its corollary), which turns many small cliques in a common neighborhood into many larger ones after dropping one vertex from the core; that properly generalizes the Fox–Wigderson dense-edge fact and is worth keeping. For s=1 they define heavy edges, show the heavy graph has bounded matching number and vertex cover F, delete F to get a K_r-free graph, refine the clique blow-up partition by neighborhoods into F, then use matching/König to force a genuine blow-up quotient that stays T-free. The r=3 appendix is clean and separate.\n\nSoft spots, in proportion. The s=1, r≥4 argument is long; the induction on overlap inside Lemma 3.4 is the load-bearing combinatorial step. I walked the cases: each either builds a forbidden T via the common-neighborhood claim or reduces overlap while preserving the hypotheses. No missing case jumped out, and the stress-test agrees. They use Liu–Shangguan–Skokan–Xu as a black box for the dense K_r-free (and neighborhood K_{r−1}-free) homomorphism bound; that is standard, and the tower in ε is allowed by the definition of δ_hom. Residual risk is the usual one for a 19-page dense counting proof, not a structural hole. Citations look appropriate; no circularity.\n\nWho cares: people working on homomorphism/chromatic thresholds, dense H-free structure, and clique-counting tools. Not a paradigm shift, but a solid exact determination plus a lemma others will cite.\n\nI would send it to referees. Engage if you work in this corner; the lemma alone is useful.","headline":"First exact homomorphism thresholds past cliques, via a clean family and a usable clique-amplification lemma; the long s=1 case is intricate but holds together.","tokens_in":20215,"tokens_out":591,"would_cite":true,"duration_ms":14207,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C35","05C15","05C60"],"pacs":[],"model":"grok-4.5","headline":"The homomorphism threshold of every clique-bundle T^s_{r,k} equals the clique value (2r-5)/(2r-3).","keywords":["homomorphism threshold","chromatic threshold","minimum degree","clique blow-ups","heavy edges","extremal graph theory","T^s_{r,k}"],"falsifier":"Exhibit a single T^s_{r,k}-free graph on n vertices with minimum degree strictly larger than ((2r-5)/(2r-3)+ε)n that admits no homomorphism into any T^s_{r,k}-free graph whose order is bounded solely in terms of r, k, s and ε.","tokens_in":20094,"feed_emoji":"🔗","tokens_out":948,"duration_ms":14082,"temperature":0.7,"pith_summary":"The paper settles the exact homomorphism threshold for an infinite family of non-complete graphs. For each r ≥ 3, k ≥ 1 and 1 ≤ s ≤ r, the graph T^s_{r,k} is formed by gluing k copies of the complete graph K_r along a common clique of size s. The authors prove that any T^s_{r,k}-free graph whose minimum degree is at least a little more than (2r-5)/(2r-3) of its order admits a homomorphism into some T^s_{r,k}-free graph of bounded size. That constant is already known to be the chromatic threshold of these graphs, so the two thresholds coincide. Until now the only graphs with a fully determined homomorphism threshold were the cliques themselves; the result therefore gives the first infinite non-clique family for which the stronger homomorphism question is answered exactly.","feed_headline":"Clique bundles share the clique homomorphism threshold","feed_subtitle":"Gluing any number of K_r’s along a common clique leaves the exact threshold (2r-5)/(2r-3) unchanged","key_machinery":"A common-neighbourhood clique-counting lemma that amplifies many small cliques in the joint neighbourhood of a tuple into many larger cliques after one vertex of the tuple is dropped. Iterating the lemma produces a bounded set of “heavy” edges whose deletion leaves a K_r-free graph to which the known clique homomorphism theorem applies.","core_discovery":"For every r ≥ 3, k ≥ 1 and 1 ≤ s ≤ r one has δ_hom(T^s_{r,k}) = (2r-5)/(2r-3). The lower bound is inherited from the known chromatic threshold; the matching upper bound is obtained by showing that every sufficiently dense maximal T^s_{r,k}-free graph is a blow-up of a bounded T^s_{r,k}-free graph.","pith_inferences":["The same amplification lemma may decide the homomorphism threshold for other graphs whose chromatic threshold equals (2r-5)/(2r-3).","The open problem posed in the paper—characterising all H with δ_hom(H)=(2r-5)/(2r-3)—is now the natural next target.","For odd cycles the gap between chromatic threshold 0 and positive homomorphism threshold remains; the new method does not immediately close it."],"forward_implications":["Gluing any number of K_r’s along a common clique of any size does not raise the homomorphism threshold above the pure-clique value.","Every graph in the family has identical chromatic and homomorphism thresholds.","The same heavy-edge and blow-up method yields an explicit structural description: dense maximal T^s_{r,k}-free graphs are blow-ups of bounded T^s_{r,k}-free quotients.","The result supplies the first infinite non-complete family whose exact homomorphism thresholds are known."],"fun_headline_variants":["Exact hom thresholds for clique bundles match clique case","T^s_{r,k} shares clique hom threshold (2r-5)/(2r-3)","Clique bundles keep the exact hom threshold of K_r","Homomorphism threshold determined beyond single cliques","Dense maximal T-free graphs are bounded blow-ups"],"cache_read_input_tokens":16512,"weakest_assumption_plain":"The argument for the hardest case relies on an external quantitative theorem that every dense maximal clique-free graph is already a blow-up of a bounded clique-free graph; if that black-box bound fails, the final bounded quotient is not guaranteed.","fun_headline_variants_meta":{"raw":{"variants":["Exact hom thresholds for clique bundles match clique case","T^s_{r,k} shares clique hom threshold (2r-5)/(2r-3)","Clique bundles keep the exact hom threshold of K_r","Homomorphism threshold determined beyond single cliques","Dense maximal T-free graphs are bounded blow-ups"]},"model":"grok-4.5","effort":"low","cost_usd":0.00445,"raw_usage":{"total_tokens":1228,"prompt_tokens":675,"num_sources_used":0,"completion_tokens":72,"cost_in_usd_ticks":44504000,"prompt_tokens_details":{"text_tokens":675,"audio_tokens":0,"image_tokens":0,"cached_tokens":128},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":481,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":675,"tokens_out":72,"duration_ms":8512,"temperature":1.0,"reasoning_tokens":481,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-31T13:34:41.007026+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Exhibit a single T^s_{r,k}-free graph on n vertices with minimum degree strictly larger than ((2r-5)/(2r-3)+ε)n that admits no homomorphism into any T^s_{r,k}-free graph whose order is bounded solely in terms of r, k, s and ε.","supporting_citations":[],"review_version":1}