{"id":"2379980b-3839-4c2d-8dcf-94dd74e2b865","arxiv_id":"2502.09576","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The authors determine the exact VC-dimension-interpolated homomorphism thresholds for cliques and prove the blowup threshold of odd cycles C_{2k-1} is 1/(2k-1).","lead":"The paper introduces a VC-dimension-aware family of degree thresholds that interpolate between chromatic and homomorphism thresholds, and exactly computes them for cliques. It also defines blowup thresholds and proves that for odd cycles C_{2k-1} the blowup threshold is 1/(2k-1), answering a question of Schacht.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.1's upper bound relies on unpublished δVC_χ(K_s) result; without a proof or published reference the bounded-coloring step in Section 3 is unsupported.","rationale":"The reader's weakest assumption is exactly the load-bearing point. Section 3 uses δVC_χ(K_s) = (s−3)/(s−2), cited to [34] ('In preparation'), to color G with C = O(1) colors before refining via Lemma 3.1. Without this bounded coloring, the auxiliary graph H can have unbounded order and the K_{s+r−3}-freeness argument does not give |H| = O(1). I checked the odd-cycle part of the paper: Theorem 1.5, Lemma 4.16, and the maximality argument for Theorem 1.3 appear internally consistent, and the Andrásfai lower bound is sound. Thus the strongest claim highlighted by the reader is not affected by this dependency. However, Theorem 1.1 is the paper's headline interpolation result, so the external dependency is significant for the paper as a whole. The CONDITIONAL verdict remains appropriate pending access to [34] or a self-contained proof.","tokens_in":34542,"tokens_out":29539,"duration_ms":226255,"concrete_test":"Obtain from the authors the proof of δVC_χ(K_s) = (s−3)/(s−2) from [34], or verify it independently: prove that every K_s-free graph with VC(G) ≤ d and δ(G) ≥ ((s−3)/(s−2) + ε)n has chromatic number at most C(ε,d,s). If this statement fails or cannot be established, Section 3's first paragraph is unsupported and Theorem 1.1's upper bound is not proven. A useful sub-check is whether the bounded coloring can be derived from Lemma 3.1 alone (e.g., by taking a = εn/2 and showing the resulting auxiliary graph has bounded order); if not, the dependency on [34] is unavoidable.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Section 3, first paragraph, the proof of Theorem 1.1's upper bound asserts that since δ(G) ≥ (β1/β0 + ε)n and δVC_χ(K_s) = (s−3)/(s−2), the graph G can be colored with a bounded number C of colors. This is the only step that turns the Haussler partition into a bounded independent partition: without a bounded chromatic number, the refinement K would not be O(1) and the auxiliary graph H would not be a bounded-size homomorphic image. The cited result is [34] (Liu, Shangguan, Xue, 'In preparation'); no proof or published version is given. If the equality δVC_χ(K_s) = (s−3)/(s−2) were false or unavailable, the bounded-coloring step collapses and the upper bound for δVC_hom(K_s;K_t) is not established. The lower-bound construction and the odd-cycle results (Theorems 1.2 and 1.3) do not use this step, so the concern is localized to Theorem 1.1 but is still load-bearing for the paper's main interpolation claim.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper introduces two new threshold parameters for H-free graphs: the bounded-VC homomorphism threshold δVC_hom(G1;G2), which interpolates between chromatic and homomorphism thresholds, and the blowup threshold δ_B(H). The main results are Theorem 1.1, computing δVC_hom(K_s;K_t)=((s-3)(t-s+2)+1)/((s-2)(t-s+2)+1); Theorem 1.2, giving two zero thresholds for families of odd cycles; Theorem 1.3, proving δ_B(C_{2k-1})=1/(2k-1); and Theorem 1.5, bounding the VC-dimension of dense maximal C_{2k-1}-free graphs. The proofs combine Haussler's packing lemma, an iterative partition-refinement process, a new inequality for codes on graphs (Theorem 1.4), and Andrásfai-graph lower-bound constructions.","tokens_in":34825,"tokens_out":23986,"duration_ms":192238,"significance":"If the results are correct, this is a substantial contribution: Theorem 1.1 gives the first smooth interpolation between chromatic and homomorphism thresholds, and Theorem 1.3 resolves the blowup threshold for all odd cycles while answering a question of Schacht. The odd-cycle arguments appear largely self-contained and introduce a useful framework for refining VC-based partitions. Theorem 1.4 is a clean statement of independent interest. The main caveat is that the upper bound of Theorem 1.1 rests on an unpublished result, and one proof in Section 4 appears to misapply a stated lemma; these issues must be resolved before the full significance can be assessed.","major_comments":[{"comment":"The proof of the upper bound of Theorem 1.1 invokes the equality δVC_χ(K_s)=(s-3)/(s-2), cited from [34] (Liu, Shangguan, Xue, 'In preparation'), to conclude that G has a bounded coloring before applying Lemma 3.1. This step is load-bearing: it is what makes the refined partition have O_{ε,d,r}(1) parts, and hence what makes the auxiliary graph H have bounded size. Because no proof or published reference is supplied, the upper bound of Theorem 1.1 is conditional on [34]. The authors should include a proof of this equality in an appendix or state Theorem 1.1 as conditional on the unpublished result.","section":"Section 3, first paragraph"},{"comment":"The first assertion of Theorem 1.2 states δVC_hom(C_{2k+1}; C_{2k-1})=0, so the input graph G is assumed only to be C_{2k+1}-free. The proof, however, applies Lemma 4.16(2), whose stated hypothesis is that G is {C_{2k+1}, C_{2k-1}}-free. Moreover, the proof of Lemma 4.16 explicitly uses C_{2k-1}-freeness of G to rule out singular C_{2k-1} cycles in H_k(G,ε/3). Thus the proof as written does not establish the stronger first assertion of Theorem 1.2. Please correct the statement of Lemma 4.16, add the missing argument, or restrict the theorem accordingly.","section":"Section 4.3, proof of Theorem 1.2"}],"minor_comments":[{"comment":"The name is misspelled as 'Obsen and Schacht'; it should be 'Ebsen and Schacht'.","section":"Section 1.1"},{"comment":"The heading 'Proof of Lemma 1.5' should refer to Theorem 1.5.","section":"Section 4.2, heading"},{"comment":"In the proof of Theorem 1.4, the symbols V(H), N_H, and H are used where the graph under discussion is G; for example, the maximum in equation (5) should be over V(G) and the neighborhoods should be in G.","section":"Section 3.2 and Claim 3.2"},{"comment":"Several LaTeX control sequences appear literally in the text, such as '/suppress Luczak' and '/suppress' before Thomassé; these should be typeset properly.","section":"Throughout"},{"comment":"Figure 1.1 is difficult to read and is not needed for the proofs; consider simplifying or removing it.","section":"Figure 1.1"}],"recommendation":"major_revision","confidential_remarks":"The paper depends heavily on [33] and especially [34], which are respectively an arXiv preprint and an in-preparation paper by overlapping author groups. This is worth monitoring for the journal's prior-publication and novelty policies, but my recommendation is based on the technical gaps described above."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This is a serious and substantial paper, and the odd-cycle result is the headline. The authors introduce two new threshold notions—bounded-VC homomorphism threshold and blowup threshold—and then actually resolve them in interesting cases. Theorem 1.3, δ_B(C_{2k−1}) = 1/(2k−1), answers Schacht's question, strengthens Ebsen–Schacht, and shows 0 is an accumulation point for blowup thresholds. That alone is a major result. Theorem 1.1, the interpolation formula for cliques, is also genuinely new: it gives a smooth curve between the chromatic and homomorphism thresholds via VC-dimension, unifying results of Thomassen, Łuczak–Thomassé, and Goddard–Lyle–Nikiforov.\n\nThe proof strategy is appealing. The lower-bound constructions in Section 2 are self-contained and the 'bad/good vertex' counting is clean. The upper bounds use Haussler's packing lemma plus an iterative refinement; the VC-dimension bound for dense maximal odd-cycle-free graphs (Theorem 1.5) is a nice use of Bollobás set-pair inequality and Ramsey.\n\nNow the soft spots, in proportion. The main one is the unpublished dependency. The upper bound of Theorem 1.1 uses δVC_χ(K_s) = (s−3)/(s−2) as a black box, cited as [34], 'In preparation,' to get a bounded coloring. If that result is not available or not correct, the bounded-coloring step in Section 3 collapses, and with it the upper bound of Theorem 1.1. The other main results, Theorems 1.2 and 1.3, do not rely on this step, so the paper's most celebrated result stands independently. But the interpolation claim is conditional on someone else's unpublished work. The authors themselves state the result in the introduction, so it's not hidden; still, a referee should insist on either a proof or a published reference.\n\nMinor issues: typos ('Obsen' for Ebsen, 'Lemma 1.5' for Theorem 1.5 in Section 4.2). The proofs are dense; I followed the main counting arguments and didn't find a gap, though I did not verify every epsilon in Lemma 4.16.\n\nWho is this for: extremal graph theorists working on thresholds, chromatic number, and VC-dimension. If the unpublished dependency gets resolved, the paper is a strong contribution. I recommend sending it to a serious referee, with instructions to focus on the use of δVC_χ(K_s) and to ask the authors to provide an appendix proof or a public preprint reference. The core is right; it is a conditional accept in the sense that Theorem 1.1 needs the missing ingredient, but the rest is solid.","headline":"A substantial paper that resolves the blowup threshold for all odd cycles and gives a genuinely new interpolation between chromatic and homomorphism thresholds, conditional only on one unpublished result used in Theorem 1.1.","tokens_in":35346,"tokens_out":4480,"would_cite":true,"duration_ms":110017,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C15","05C35"],"pacs":[],"model":"deepseek-v4-flash","headline":"Odd cycles have blowup threshold 1/(2k−1), and clique thresholds interpolate smoothly via VC-dimension.","keywords":["chromatic threshold","homomorphism threshold","blowup threshold","VC-dimension","odd cycles","clique-free graphs","Andrásfai graphs","extremal graph theory"],"falsifier":"For k = 2, search for a maximal C_5-free graph with minimum degree at least (1/5 + ε)n for some ε > 0 that is not a blowup of any bounded-size graph; Theorem 1.3 predicts none exists, so a single example would refute it. Independently, verify the value δVC_χ(K_4) = 2/3; a different value would break the upper-bound argument of Theorem 1.1.","tokens_in":34369,"feed_emoji":"🔄","tokens_out":7429,"duration_ms":56695,"temperature":0.7,"pith_summary":"This paper introduces a generalized threshold that interpolates chromatic and homomorphism thresholds for graphs with bounded VC-dimension, and it determines this threshold exactly for all pairs of cliques. In the same framework, it introduces the blowup threshold δ_B(H), which asks for the minimum degree fraction that forces a maximal H-free graph to be a blowup of a bounded-size graph. The paper's central result is that for odd cycles, δ_B(C_{2k−1}) = 1/(2k−1) for every integer k ≥ 2. This means every maximal C_{2k−1}-free graph on n vertices with minimum degree at least (1/(2k−1)+ε)n is a blowup of a graph of constant size, while at the exact threshold there are twin-free examples that are not. The results answer a question of Schacht and show that blowup thresholds, unlike chromatic thresholds, have 0 as an accumulation point.","feed_headline":"Blowup threshold for odd cycles is exactly 1/(2k-1)","feed_subtitle":"Every maximal odd-cycle-free graph with a bit more minimum degree is a bounded blowup; the bound is sharp.","key_machinery":"The arguments run on two engines. The first is the theory of VC-dimension: a Haussler packing lemma (Lemma 3.1) partitions the vertex set into classes with nearly identical neighborhoods, and an iterative refinement (Definition 4.13) sharpens this partition until each class is either complete or anti-complete to every other class, yielding the blowup structure. The second is a code-on-graphs theorem (Theorem 1.4), which bounds the total ℓ1-weight of binary vectors assigned to vertices of a dense graph under a constraint on common 1-coordinates of any K_{s−2}; this theorem supplies the clique-number bound on the homomorphic image in Theorem 1.1. For the odd-cycle results, Bollobás's set-pair inequality bounds the VC-dimension of dense maximal C_{2k−1}-free graphs (Theorem 1.5), and Andrásfai graphs provide the sharp lower-bound constructions.","core_discovery":"The paper proves two main structural claims. First, for integers t ≥ s ≥ 3, the bounded-VC homomorphism threshold for mapping K_s-free graphs to K_t-free images is δVC_hom(K_s;K_t) = ((s−3)(t−s+2)+1)/((s−2)(t−s+2)+1), with matching constructions showing optimality. As t grows this value descends smoothly from δVC_hom(K_s) = (2s−5)/(2s−3) to δVC_χ(K_s) = (s−3)/(s−2), showing that the coincidence of chromatic and homomorphism thresholds for cliques is governed by VC-dimension. Second, the paper defines the blowup threshold δ_B(H) and proves δ_B(C_{2k−1}) = 1/(2k−1) for all k ≥ 2: any maximal C_{2k−1}-free graph with minimum degree above (1/(2k−1)+ε)n is a blowup of a graph whose size is bounded in terms of k and ε, and the Andrásfai graphs show the threshold cannot be lowered. Along the way the paper proves that such graphs have bounded VC-dimension and that removing O_{ε,k}(1) vertices destroys all short odd cycles.","pith_inferences":["The paper does not compute δ_B for other graphs; testing Conjecture 5.1 (δ_hom = δ_B) on even cycles or on K_{s,t} would be a natural next step, and Theorem 1.4's code bound may be the right tool.","The iterative refinement that forces blowup structure suggests a general sufficient condition: any H-free graph whose VC-dimension is bounded and whose minimum degree is a positive fraction of n should be near a blowup, potentially extending beyond odd cycles.","Theorem 1.2's zero threshold for {C_{2k+1}, C_{2k−1}}-free graphs suggests the blowup threshold of the family {C_{2k+1}, C_{2k−1}} might also be 0; verifying or refuting this would be a concrete test of the framework.","The paper leaves open whether δVC(C_{2k−1}) equals 1/(2k−1); resolving this would pin down when bounded VC-dimension alone, rather than maximality, forces blowup structure."],"forward_implications":["For every k ≥ 2, every maximal C_{2k−1}-free graph with minimum degree at least (1/(2k−1)+ε)n is a blowup of a graph of size bounded by a tower function in k and 1/ε.","The Andrásfai graphs show the threshold is sharp: at minimum degree n/(2k−1) there are maximal C_{2k−1}-free graphs that are twin-free and hence not blowups of any smaller graph.","Since 1/(2k−1) → 0 as k → ∞, blowup thresholds have 0 as an accumulation point, in contrast to chromatic thresholds, which are all jumps.","The interpolation formula for cliques, δVC_hom(K_s;K_t) = ((s−3)(t−s+2)+1)/((s−2)(t−s+2)+1), is exact and decreases to δVC_χ(K_s) as t → ∞.","For mixed odd-cycle families, δVC_hom(C_{2k+1}; C_{2k−1}) = 0, extending the VC-dimension transition for homomorphism thresholds from 1/(2k+1) to 0."],"supporting_citations":[{"why":"Establishes the upper bound δ_hom(C_{2k−1}) ≤ 1/(2k−1) that Theorem 1.3 strengthens from homomorphism to blowup structure.","marker":"[14]"},{"why":"Provides the structural analysis and induced-6-cycle lemma for C_5-free graphs used in the warm-up proof of Theorem 4.2.","marker":"[31]"},{"why":"Gives the first separation between chromatic and homomorphism thresholds for odd cycles, which the paper complements with the blowup threshold.","marker":"[42]"},{"why":"Supplies the partition lemma (Lemma 3.1) and the observation that bounded VC-dimension does not lower homomorphism thresholds for cliques.","marker":"[33]"},{"why":"Provides the unpublished value δVC_χ(K_s) = (s−3)/(s−2) that the upper-bound proof of Theorem 1.1 assumes.","marker":"[34]"},{"why":"Haussler's packing bound, the basis for the partitions used throughout the paper.","marker":"[29]"},{"why":"Bollobás's set-pair inequality, used to bound the VC-dimension of dense maximal odd-cycle-free graphs in Theorem 1.5.","marker":"[7]"}],"fun_headline_variants":["Odd cycle blowup threshold: exactly 1/(2k-1)","VC-dimension unifies chromatic and homomorphism thresholds","New thresholds interpolate graph coloring and homomorphism","Blowup threshold settles odd cycle case at 1/(2k-1)"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The upper bound for the clique interpolation theorem assumes the still-unpublished result δVC_χ(K_s) = (s−3)/(s−2); if that result were false, the bounded-coloring step in Section 3 would collapse, though Theorems 1.2 and 1.3 do not depend on it.","fun_headline_variants_meta":{"raw":{"variants":["Odd cycle blowup threshold: exactly 1/(2k-1)","VC-dimension unifies chromatic and homomorphism thresholds","New thresholds interpolate graph coloring and homomorphism","Blowup threshold settles odd cycle case at 1/(2k-1)"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000217,"raw_usage":{"total_tokens":1591,"prompt_tokens":1253,"completion_tokens":338,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":869,"completion_tokens_details":{"reasoning_tokens":266}},"tokens_in":869,"tokens_out":338,"duration_ms":3676,"temperature":1.0,"reasoning_tokens":266,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T20:58:06.608337+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For k = 2, search for a maximal C_5-free graph with minimum degree at least (1/5 + ε)n for some ε > 0 that is not a blowup of any bounded-size graph; Theorem 1.3 predicts none exists, so a single example would refute it. Independently, verify the value δVC_χ(K_4) = 2/3; a different value would break the upper-bound argument of Theorem 1.1.","supporting_citations":[{"cited_title":"Ebsen and M","cited_arxiv_id":null,"evidence_quote":"Establishes the upper bound δ_hom(C_{2k−1}) ≤ 1/(2k−1) that Theorem 1.3 strengthens from homomorphism to blowup structure."},{"cited_title":"Letzter and R","cited_arxiv_id":null,"evidence_quote":"Provides the structural analysis and induced-6-cycle lemma for C_5-free graphs used in the warm-up proof of Theorem 4.2."},{"cited_title":"Homotopy and the Homomorphism Threshold of Odd Cycles","cited_arxiv_id":"2206.07525","evidence_quote":"Gives the first separation between chromatic and homomorphism thresholds for odd cycles, which the paper complements with the blowup threshold."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the unpublished value δVC_χ(K_s) = (s−3)/(s−2) that the upper-bound proof of Theorem 1.1 assumes."},{"cited_title":"Haussler","cited_arxiv_id":null,"evidence_quote":"Haussler's packing bound, the basis for the partitions used throughout the paper."},{"cited_title":"Bollob´ as","cited_arxiv_id":null,"evidence_quote":"Bollobás's set-pair inequality, used to bound the VC-dimension of dense maximal odd-cycle-free graphs in Theorem 1.5."}],"review_version":1}