{"id":"8f23412e-607e-4653-ac77-793aa5919c7c","arxiv_id":"2506.09020","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Every rational exponent in (1,2) is realized by a family of at most 2^a induced forbidden bipartite graphs in K_{s,s}-free hosts, with new optimal induced bounds for theta and prism graphs.","lead":"This paper proves that many classical extremal results for bipartite graphs can be transferred to the setting where the forbidden copies are induced, as long as the host graph contains no complete bipartite subgraph K_{s,s}. It realizes every rational number between 1 and 2 as an induced Turán exponent with a small forbidden family, and gives new optimal-order bounds for induced theta and prism graphs.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Section 4.2's claim that each degenerate 2ℓ-cycle is counted in at most one A_{u,v} is false: figure-eight walks are counted under multiple rotations, so inequality (2) needs a factor of ℓ, which the very loose constants can likely absorb.","rationale":"The reader's weakest-assumption pinpoints a real false statement in the proof of Theorem 4.2. I agree that this is the most load-bearing concern: Theorem 4.2 is the engine behind the induced theta-graph upper bound (Theorem 1.5), one of the paper's three advertised optimal bounds. The claimed uniqueness in §4.2 is not a minor typo—it is used to control the total red degree before averaging—but the failure mode is bounded (each degenerate walk has only 2ℓ rotations, so multiplicity ≤2ℓ) and the proof's constants are exponential in ℓ and in (ℓt)^{20s}, leaving ample room to absorb an extra factor of ℓ. I also noted two smaller issues that do not change the picture: in Theorem 1.3 the greedy set V_k does not explicitly exclude previously embedded vertices, so the count can include non-injective maps (fix: remove the ≤t used vertices; d/(2K)≫t makes this harmless), and the '2^a' family-size bound in Theorem 1.2 is stated with the numerator of q rather than of 2−q (the construction gives the latter, which is smaller, so the theorem remains true). Neither affects the central claims. The prism proof (Section 5) is long but internally consistent, and the lower-bound blowup construction is sound. Therefore the appropriate verdict is unchanged: conditional acceptance pending a corrected counting argument in §4.2.","tokens_in":22276,"tokens_out":55527,"duration_ms":583946,"concrete_test":"Exhibit the ℓ=3 closed walk W = a-b-c-a-d-c-a and list all rotations (u,v) for which the two length-3 paths share an interior vertex; confirm W lies in at least two distinct sets A_{u,v}, disproving the 'at most one' assertion. Then re-run the proof of Theorem 4.2 with (2) replaced by Σ|A_{u,v}| ≤ 2ℓ·#{degenerate homomorphic 2ℓ-cycles}. Verify that with C=(ℓt)^{100ℓ} (or larger) the combined inequality still yields a pair (u*,v*) satisfying |A_{u*,v*}| ≪ |P_{u*,v*}|^2, |B_{u*,v*}| ≥ ε/(16ℓ)|P_{u*,v*}|^2, and |P_{u*,v*}| ≥ (C f(s))^ℓ·sqrt(ε/(16ℓ)), so that Lemma 4.4 applies and the final K_{s,s} contradiction is unchanged.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 4.2 upper-bounds Σ(u,v)|A_{u,v}| by the number of degenerate homomorphic 2ℓ-cycles via the assertion that each degenerate cycle is counted in at most one set A_{u,v}. This is false. A closed walk consisting of two ℓ-cycles sharing a single vertex x (e.g., ℓ=3: a-b-c-a-d-c-a) is degenerate, and for each rotation starting at a_i on one cycle, the two ℓ-paths from a_i to b_i share x, so the walk is counted in several distinct sets A_{u,v}. Hence the right-hand side of (2) is off by a factor up to 2ℓ. This is load-bearing because (2) is used to control the total red edge count in the averaging step that produces (u*,v*). The error is likely repairable: the proof only needs an upper bound on Σ|A_{u,v}|, and replacing the factor 1 by 2ℓ can be absorbed by rescaling the coefficient of Σ|A| in the combined inequality. Since C>(ℓt)^{100ℓ} is enormous, one can divide the offending coefficient by ℓ and still obtain the required lower bound on |B_{u*,v*}| relative to |A_{u*,v*}| and |P_{u*,v*}|, so Lemma 4.4 still applies and the final K_{s,s} contradiction goes through with constants adjusted.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies induced Turán numbers under a K_{s,s}-freeness assumption on the host graph, proposing a transfer phenomenon from non-induced to induced extremal problems. The main results are: (i) every rational exponent q in (1,2) is realized as the induced Turán exponent of a subfamily of the Bukh–Conlon family of size at most 2^a (Theorem 1.2); (ii) supersaturation theorems for induced trees and induced even cycles in K_{s,s}-free almost-regular graphs (Theorems 1.3, 4.2, 5.2); and (iii) optimal-order upper bounds for induced theta graphs and induced even prisms (Theorems 1.5 and 1.6), matching the known non-induced exponents. The lower bounds are obtained by clique blowups of known constructions, while the upper bounds are proved by passing to almost-regular subgraphs and using auxiliary Ramsey and dependent-random-choice arguments.","tokens_in":22574,"tokens_out":55767,"duration_ms":641374,"significance":"If the proofs are correct, the paper gives substantial new evidence for the Hunter–Milojević–Sudakov–Tomon conjecture and for the proposed transfer phenomenon. The induced-rational-exponent result improves the Bukh–Conlon family size from an uncontrolled number to at most 2^a, and the induced theta and prism bounds match the best known non-induced exponents. The paper is largely self-contained: it supplies new supersaturation machinery, explicit constants, and a clean lower-bound construction via clique blowups. These are genuine contributions. The two technical gaps I identify below are localized and appear repairable within the paper's existing framework, so the high-level claims are plausible despite the need for revision.","major_comments":[{"comment":"The assertion before (2) that every degenerate homomorphic 2ℓ-cycle is counted in at most one set A_{u,v} is false. For ℓ=3, the closed walk a-b-c-a-d-c-a is degenerate, and it is counted in A_{b,d} when read starting at b with midpoint d and in A_{d,b} when read starting at d; the two ℓ-paths share the interior vertex c in both representations. More generally, a figure-eight walk can be counted under several rotations, so the left side of (2) should be bounded by O(ℓ) times the number of degenerate homomorphic cycles, not by that number itself. This error is load-bearing: with the uncorrected coefficient, the displayed inequality before the averaging step is no longer valid once the extra factor from (2) is inserted, and the derivation of the pair (u*,v*) is unsupported. The argument is repairable: replacing the coefficient of Σ|A| by its 1/(2ℓ) multiple (equivalently, enlarging the parameter δ in the application of Lemma 4.4 by a factor 2ℓ) is harmless because C>(ℓt)^{100ℓ} leaves enormous slack. As written, however, the proof of (2) and the subsequent averaging step are incorrect.","section":"Section 4.2, Eq. (2)"},{"comment":"The counting of embeddings does not exclude previously embedded vertices from the candidate set V_k. If w_i is an earlier leaf of the prefix tree whose only earlier neighbor is w_{k'}, then v_i lies in N(v_{k'}), is not in N(v_j) for any j≠k', and need not belong to any X(v_j); hence v_i is counted as an available choice for v_k even though reusing v_i would make the map non-injective. The lower bound |V_k|≥d/(2K) therefore overcounts valid embeddings. This gap is also repairable: subtracting at most t previously embedded vertices from each candidate set is absorbed by the slack in the assumption d≥(4Kt)^{6s}s^3, and the final constant in Theorem 1.3 would still have the required form. As stated, however, the proof does not justify the claimed number of induced copies.","section":"Section 3.2, proof of Theorem 1.3"}],"minor_comments":[{"comment":"Displayed results are referred to by inconsistent labels: Lemma 2.1 is called Theorem 2.1 in its proof; Lemma 3.3 is cited as Theorem 3.3; Claim 3.4 as Theorem 3.4; Lemmas 4.3, 4.4, 5.3, 5.4, 5.5, 5.9 and Claim 5.10 are cited as Theorems 4.3, 4.4, 5.3, 5.4, 5.5, 5.9, 5.10; and Proposition 1.4 is called Theorem 1.4 in Sections 4 and 5. These labels should be reconciled.","section":"Throughout"},{"comment":"The text states that Lemma 4.4 yields 'at least δr²/2 = εr²/(64ℓ) pairs' among the selected paths, but the quantity used in Lemma 4.4 is the blue-edge parameter λ=ε/(32ℓ), not δ; the displayed equality is correct only if 'δ' is replaced by 'λ'.","section":"Section 4.2, after Lemma 4.4"},{"comment":"In the displayed lower bound for the number of induced copies of (T;R) under a fixed root orientation, the final term '≥ C/(4K)' appears to be missing the exponent and the cancellation of the powers of n; as written the algebra is unclear. The intended estimate should be that the ratio is at least (C/(4K))^{t-1}, since the exponent of n is 0 after substituting α=1-x/y and y=t-1.","section":"Section 3.1, proof of Theorem 3.1"},{"comment":"The notation for rational exponents is inconsistent: Theorem 1.2 writes q=a/b with q∈(1,2), while the remark in Section 3.2 discusses 'a/b∈(0,1)' as the deficit from 2 and uses rooted trees of density b/a. The change of variables should be stated explicitly to avoid confusion.","section":"Section 1.1 and Section 3.2"},{"comment":"The claim that at most 4 of the 2ℓ rotations of a homomorphic 2ℓ-cycle are special is asserted without proof; a one-sentence explanation of why the forbidden positions {1,ℓ+1} account for at most four rotations per edge would improve readability.","section":"Section 5.3, Claim 5.6"}],"recommendation":"major_revision","confidential_remarks":"The two technical gaps are localized and, in my assessment, both are repairable with the constants already present in the paper (the factor ℓ in Section 4.2 and the at-most-t collision issue in Section 3.2 are both absorbed by the very large slack in C and d). I therefore recommend inviting a revision rather than rejecting. There is no indication of a novelty or attribution problem."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Straight to the point: this is a substantial paper and the main results look like they belong in a good combinatorics journal. The one thing to check before you cite it is Section 4.2: the claim that each degenerate homomorphic 2ℓ-cycle is counted in at most one A_{u,v} is false. A figure-eight closed walk (two ℓ-cycles sharing a vertex) is counted under several rotations in different A_{u,v}; I'd expect a factor of up to 2ℓ in inequality (2). The good news is that this is repairable: the paper's constants are so loose (C > (ℓt)^{100ℓ}) that multiplying the RHS by 2ℓ and adjusting the averaging step still leaves enough room for Lemma 4.4 to apply. So I view this as a fixable gap, not a fatal flaw.\n\nWhat the paper does well: it proves a genuine transfer principle. Theorem 1.2 realizes every rational exponent q∈(1,2) as an induced Turán exponent with a forbidden family that is a subfamily of the Bukh–Conlon family, substantially smaller than the trivial 'all supergraphs' family. The upper bound depends on an induced-tree supersaturation theorem (Theorem 1.3) that is optimal in the exponent, plus a Ramsey-type argument to clean up messy intersections. The theta and prism results (Theorems 1.5, 1.6) give optimal n-dependencies and support the Hunter–Milojević–Sudakov–Tomon conjecture; the proofs are detailed and the authors are careful about the dependence on s, even if the upper bounds are exponential in s. I also like that they proved a self-contained induced C4 supersaturation result (Theorem 5.2) rather than hiding behind a citation.\n\nSoft spots, in order of severity. First, the Section 4.2 issue I mentioned; it is localized and probably fixable with a factor of 2ℓ and a constant adjustment. Second, the paper systematically omits floors and ceilings; in the blowup construction (Proposition 1.4) and the almost-regularity lemma (Theorem 2.1) this is standard but should be clarified. Third, the prism result stops at ℓ≥10 and the authors say why (the proof of Claim 5.7); that's a real limitation for the conjecture but not a flaw in the current theorem. Fourth, the lower bounds in Theorems 1.5 and 1.6 have polynomial dependence on s while the upper bounds are exponential; the authors acknowledge this in the conclusions.\n\nWho is this for: researchers working on bipartite Turán problems, induced subgraphs, or the HMST conjecture. It deserves a serious referee; I'd send it to a journal like Combinatorica or JCTA. If I were the editor, I'd ask the authors to fix the Section 4.2 counting and resubmit.","headline":"Strong transfer results in induced Turán theory; one repairable counting error in Section 4.2.","tokens_in":23103,"tokens_out":7781,"would_cite":true,"duration_ms":85699,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C35","05C38"],"pacs":[],"model":"deepseek-v4-flash","headline":"Every rational exponent in (1,2) is realized by an induced Turán family of size at most $2^a$ in $K_{s,s}$-free host graphs.","keywords":["induced Turán numbers","rational exponents","K_{s,s}-free graphs","supersaturation","theta graphs","prism graphs","induced trees","even cycles"],"falsifier":"Take the degenerate closed walk formed by two internally vertex-disjoint $\\ell$-cycles meeting at exactly one vertex, and count the ordered pairs $(u,v)$ of vertices at distance $\\ell$ along the walk: the walk lies in several distinct sets $A_{u,v}$, demonstrating that the inequality $\\sum_{(u,v)\\in\\Gamma}|A_{u,v}|\\le \\#\\{\\text{degenerate $2\\ell$-cycles}\\}$ fails by a factor of at least $\\ell$; checking whether the corrected factor still fits inside the slack constants of the proof settles whether Theorem 4.2 stands as written.","tokens_in":22070,"feed_emoji":"📈","tokens_out":6269,"duration_ms":75670,"temperature":0.7,"pith_summary":"This paper claims a broad transfer phenomenon: many extremal results for bipartite graphs remain true when the forbidden condition is induced, provided the host graph is $K_{s,s}$-free. Concretely, for every rational $q=\\frac{a}{b}\\in(1,2)$, the paper constructs a family $\\mathcal{H}$ of at most $2^a$ bipartite graphs whose induced Turán number is $\\Theta_s(n^q)$, extending the Bukh–Conlon rational-exponent theorem to the induced setting. The engine is an optimal supersaturation result showing that dense, almost-regular $K_{s,s}$-free graphs contain many induced copies of every fixed tree, and analogous supersaturation for even cycles. Using the same machinery, the paper proves optimal-order induced Turán bounds for $\\theta$ graphs and for even prism graphs, matching their known ordinary Turán exponents and giving strong evidence for the Hunter–Milojević–Sudakov–Tomon conjecture. A reader should care because the transfer makes a large body of bipartite extremal knowledge available to induced problems under a mild local-sparsity condition.","feed_headline":"Induced Turán numbers hit every rational exponent from 1 to 2","feed_subtitle":"Each exponent comes from at most 2^a induced forbidden graphs, with matching bounds for theta and prism graphs.","key_machinery":"The central objects are the $(p;S)$-lifts $F_p(T;R)$ of a rooted tree $(T;R)$: graphs obtained by gluing $p$ vertex-disjoint copies of $T$ along the root set $R$ and an additional set $S$, where $S$ ranges over subsets of the non-root vertices. The paper proves a Ramsey-type lemma that, from many induced copies of $T$ sharing a fixed root set, extracts $p$ copies whose intersections are regular and whose union is induced, forming a lift in $F_p(T;R)$. The extraction is powered by two ingredients: a greedy embedding that uses $K_{s,s}$-freeness to control bad common neighborhoods, and an almost-regular reduction (Lemma 2.1) that keeps degrees and codegrees bounded. For the cycle-based theorems, the machinery combines Sidorenko's property for even cycles, the Kővári–Sós–Turán theorem, and Janzer's regularization lemma to show that most closed $2\\ell$-walks in a dense $K_{s,s}$-free graph are induced $2\\ell$-cycles, from which induced $\\theta$ and prism subgraphs are assembled.","core_discovery":"The central discovery is that the induced Turán number $\\mathrm{ex}^*(n,\\mathcal{H},s)$ — the maximum number of edges in an $n$-vertex $K_{s,s}$-free graph with no induced copy of any graph in $\\mathcal{H}$ — satisfies the same rational exponents as the ordinary Turán number for a carefully chosen subfamily of the Bukh–Conlon lifting family. For every rational $q=\\frac{a}{b}\\in(1,2)$, the paper finds a family $\\mathcal{H}$ of at most $2^a$ bipartite graphs with $\\mathrm{ex}^*(n,\\mathcal{H},s)=\\Theta_s(n^q)$ (Theorem 1.2). It also proves that for $\\theta$ graphs $\\Theta^t_{\\ell}$, $\\mathrm{ex}^*(n,\\Theta^t_{\\ell},s)=\\Theta_{\\ell,t}(n^{1+1/\\ell})$ up to $s$-dependent constants for all sufficiently large $t$ (Theorem 1.5), and that for even prism graphs $C^{\\square}_{2\\ell}$ with $\\ell\\ge 10$, $\\mathrm{ex}^*(n,C^{\\square}_{2\\ell},s)=\\Theta_{\\ell}(n^{3/2})$ (Theorem 1.6). In both cases the $n$-exponent matches the ordinary Turán exponent, thereby confirming the transfer phenomenon for these families.","pith_inferences":["The double-counting gap in Section 4.2, where a degenerate closed walk formed by two cycles sharing a single vertex is counted in several sets $A_{u,v}$, is most likely absorbed by the very loose constants in the proof; a corrected factor of $\\ell$ should not change the theorem's truth.","The transfer phenomenon probably extends to any bipartite $H$ whose ordinary Turán exponent is realized by lifted-tree constructions, as long as an induced-supersaturation statement holds for the building blocks; the paper's tree and even-cycle supersaturation results are natural templates.","The exponential $s$-dependence in the upper bounds for theta and prism graphs, contrasted with polynomial lower bounds, is an artifact of the method; closing this gap is a test of whether the transfer is tight in the parameters beyond $n$.","A further reduction from the $2^a$-element family to a single forbidden graph would settle Conjecture 1.1 but requires a fundamentally new idea, since the Ramsey cleaning step inherently needs many copies to find a regular intersection."],"forward_implications":["Every rational number in $(1,2)$ is the induced Turán exponent of a family of at most $2^a$ bipartite graphs, with the family size independent of $s$ and of $b$.","For theta graphs, the induced Turán number has the same $n$-exponent as the ordinary Turán number, $n^{1+1/\\ell}$, for all sufficiently large $t$; for even prisms with $\\ell\\ge 10$, the induced Turán number is $\\Theta_{\\ell}(n^{3/2})$.","The supersaturation theorem shows that any $K$-almost-regular $K_{s,s}$-free graph with average degree $d$ contains at least $n(d/2K)^{t-1}$ labeled induced copies of every $t$-vertex tree, which is optimal up to constants.","The transfer phenomenon implies that ordinary bipartite Turán results can be lifted to induced statements in locally sparse hosts, making the Hunter–Milojević–Sudakov–Tomon conjecture plausible for a wider class of bipartite graphs.","The clique-blowup construction (Proposition 1.4) converts any ordinary lower bound $\\mathrm{ex}(n,H)=\\Omega(n^{1+\\alpha})$ into an induced lower bound $\\Omega_h(s^{1-\\alpha}n^{1+\\alpha})$, so for connected bipartite $H$ the induced Turán number is never much smaller than the ordinary one."],"supporting_citations":[{"why":"Supplies the lifting families and the rational-exponent framework that Theorem 1.2 extends to the induced setting, including the lower-bound constructions from its Section 2.3.","marker":"[7]"},{"why":"Introduces the conjecture that $\\mathrm{ex}^*(n,H,s)=O_{H,s}(\\mathrm{ex}(n,H))$ and provides the structural lemmas (Lemma 4.3 and 4.4) used in the theta upper bound.","marker":"[28]"},{"why":"Determines the ordinary Turán exponent of theta graphs, which the induced lower bound of Theorem 1.5 relies on via the transfer proposition.","marker":"[10]"},{"why":"Establishes $\\mathrm{ex}(n,C^{\\square}_{2\\ell})=\\Theta_{\\ell}(n^{3/2})$ for $\\ell\\ge 4$; its thick/thin 4-cycle decomposition strategy is adapted to the induced prism proof.","marker":"[23]"},{"why":"The Kővári–Sós–Turán bound is used throughout to control codegrees and edge densities in $K_{s,s}$-free graphs, underpinning the bad-neighborhood and regularization arguments.","marker":"[33]"},{"why":"Provides the bipartite regularization and homomorphism-counting lemmas (Lemmas 2.2–2.4) that control degenerate closed walks in the prism argument.","marker":"[30]"},{"why":"Sidorenko's property for even cycles supplies the lower bound on homomorphisms of $C_{2\\ell}$ used both in the theta and prism supersaturation arguments.","marker":"[41]"}],"fun_headline_variants":["Small induced families realize every rational Turan exponent in (1,2)","All rational Turan exponents from 1 to 2 via ≤2^a induced graphs","Every rational Turan exponent in (1,2) from ≤2^a induced forbidden graphs","Induced Turan numbers realize all rational exponents in (1,2) via ≤2^a graphs"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof of Theorem 4.2 assumes that every degenerate homomorphic $2\\ell$-cycle is counted in at most one of the sets $A_{u,v}$; a closed walk consisting of two cycles sharing a single vertex is counted in several such sets under different rotations, so the asserted uniqueness is false and the argument needs an extra factor of $\\ell$.","fun_headline_variants_meta":{"raw":{"variants":["Small induced families realize every rational Turan exponent in (1,2)","All rational Turan exponents from 1 to 2 via ≤2^a induced graphs","Every rational Turan exponent in (1,2) from ≤2^a induced forbidden graphs","Induced Turan numbers realize all rational exponents in (1,2) via ≤2^a graphs"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.002569,"raw_usage":{"total_tokens":9907,"prompt_tokens":1085,"completion_tokens":8822,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":701,"completion_tokens_details":{"reasoning_tokens":8727}},"tokens_in":701,"tokens_out":8822,"duration_ms":72932,"temperature":1.0,"reasoning_tokens":8727,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T05:01:48.610047+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the degenerate closed walk formed by two internally vertex-disjoint $\\ell$-cycles meeting at exactly one vertex, and count the ordered pairs $(u,v)$ of vertices at distance $\\ell$ along the walk: the walk lies in several distinct sets $A_{u,v}$, demonstrating that the inequality $\\sum_{(u,v)\\in\\Gamma}|A_{u,v}|\\le \\#\\{\\text{degenerate $2\\ell$-cycles}\\}$ fails by a factor of at least $\\ell$; checking whether the corrected factor still fits inside the slack constants of the proof settles whether Theorem 4.2 stands as written.","supporting_citations":[{"cited_title":"Bukh and D","cited_arxiv_id":null,"evidence_quote":"Supplies the lifting families and the rational-exponent framework that Theorem 1.2 extends to the induced setting, including the lower-bound constructions from its Section 2.3."},{"cited_title":"Hunter, A","cited_arxiv_id":null,"evidence_quote":"Introduces the conjecture that $\\mathrm{ex}^*(n,H,s)=O_{H,s}(\\mathrm{ex}(n,H))$ and provides the structural lemmas (Lemma 4.3 and 4.4) used in the theta upper bound."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Determines the ordinary Turán exponent of theta graphs, which the induced lower bound of Theorem 1.5 relies on via the transfer proposition."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes $\\mathrm{ex}(n,C^{\\square}_{2\\ell})=\\Theta_{\\ell}(n^{3/2})$ for $\\ell\\ge 4$; its thick/thin 4-cycle decomposition strategy is adapted to the induced prism proof."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the bipartite regularization and homomorphism-counting lemmas (Lemmas 2.2–2.4) that control degenerate closed walks in the prism argument."},{"cited_title":"Sidorenko","cited_arxiv_id":null,"evidence_quote":"Sidorenko's property for even cycles supplies the lower bound on homomorphisms of $C_{2\\ell}$ used both in the theta and prism supersaturation arguments."}],"review_version":1}