{"id":"de5ebcce-e487-4684-acc8-8ba99edbcb8e","arxiv_id":"2507.23375","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":0.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A review of recent advances in arrow relations and traces of sets, presenting known theorems, constructions, and open problems without new results.","lead":"This paper surveys recent results on arrow relations and traces of finite set families, a core topic in extremal set theory. It organizes the field into defect Sauer problems, trace functions, extremal families, forbidden matrix configurations, and Turán-type trace problems.","discovery_kind":"review","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Internal inconsistency in the proof of Theorem 3 (Frankl's m(n,3,7) result): a hereditary family with |F| = floor(n^2/4)+n+2 and F ↛ (3,7) is claimed impossible, but the displayed extremal lower-bound family has size floor(n^2/4)+n+1, exactly one less, so the contradiction and the stated upper…","rationale":"The paper is a survey, so the most valuable contribution is accurate transmission of known results. The reader identified a citation error (Hajnal vs Bondy) in the introduction, which is a real fidelity problem. My stress-test found a more serious issue: the proof of Theorem 3 in Section 3 contains an internally inconsistent arithmetic inequality. The proof claims to derive a contradiction from |F| = floor(n^2/4)+n+2 by bounding |F| by 1+n+ex(n,K3) = 1+n+floor(n^2/4), which is arithmetically impossible - it would imply 2+n+floor(n^2/4) ≤ 1+n+floor(n^2/4). The lower-bound construction given immediately after has size floor(n^2/4)+n+1, which is one less than the value in the theorem. This suggests either a misstatement of the theorem (the intended value might be floor(n^2/4)+n+1) or a gap in the proof as reproduced. Because the survey's value depends on accurate reproduction of proofs and results, this inconsistency is load-bearing: a reader cannot verify the theorem from the given proof, and if the value is wrong, the table and subsequent discussion inherit the error. However, I have not checked the original literature to determine which side is correct, so I do not change the verdict from CONDITIONAL; I recommend the same conditional acceptance, but with a specific request to fix this proof. The reader's weakest assumption (fidelity to primary literature) is the same category of concern, so we partially agree, but the specific technical failure I identified is distinct from the Hajnal/Bondy citation error.","tokens_in":22706,"tokens_out":2480,"duration_ms":22180,"concrete_test":"Recompute the lower-bound construction F = C([n],0) ∪ C([n],1) ∪ E(T(n,2)): its size is 1 + n + floor(n^2/4). Check whether this family is hereditary (if it contains all singletons and the empty set, it is hereditary) and check that no 3-set has a trace of size 7. If it is hereditary and |F| = floor(n^2/4)+n+1, then the proof's contradiction at |F| = floor(n^2/4)+n+2 is invalid as written: the survey's displayed equality 2+n+floor(n^2/4) ≤ 1+n+floor(n^2/4) is arithmetically false. Re-derive the intended upper bound: perhaps the intended hereditary family should omit the empty set, or the statement should be m(n,3,7) = floor(n^2/4)+n+1, or the inequality in the proof should be |F| = 2+n+floor(n^2/4) ≤ 1+n+ex(n,K3) = 1+n+floor(n^2/4), which is impossible.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The decisive step in Section 3's proof of Theorem 3 is the upper bound. The authors suppose m(n,3,7) > floor(n^2/4)+n+2 and, by Lemma 1, take a hereditary F with |F| = floor(n^2/4)+n+2 and F ↛ (3,7). They then note that if F contains a 3-element set, that set is shattered, giving a trace of size 8, so all sets in F have size at most 2. They also reason that if the 2-element sets contain a triangle {a,b},{a,c},{b,c}, then |F|{a,b,c}| ≥ 7, contradicting F ↛ (3,7). Hence the 2-sets form a triangle-free graph, so by Mantel the number of 2-sets is at most floor(n^2/4). Adding the empty set and all n singletons (which hereditary F must contain if it contains any 2-set) gives |F| ≤ 1+n+floor(n^2/4), which does not contradict |F| = floor(n^2/4)+n+2. The displayed inequality in the paper actually reads 2+n+floor(n^2/4) = |F| ≤ 1+n+ex(n,K3) = 1+n+floor(n^2/4), which is arithmetically impossible. So either the extremal construction below is missing the empty set or a singleton, or the target value in Theorem 3 should be floor(n^2/4)+n+1. This is a concrete internal inconsistency in the survey's reproduction of a foundational proof; a reader cannot verify the stated theorem from the given argument.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper is a survey of arrow relations and traces of finite sets. It organizes the literature into defect Sauer results, single-element removal, trace functions, extremal families, forbidden configurations in matrices, and Turán numbers for traces, and it closes with open problems. The survey introduces no new theorems; its contribution is a unified presentation of known results, including reproduced proofs of Frankl's hereditary-family lemma (Lemma 1), the determination of m(n,3,7) (Theorem 3), and Frankl's lower bound for m(n,2^{d-1}-1) (Theorem 8), together with tables of known values and a substantial bibliography.","tokens_in":23091,"tokens_out":23946,"duration_ms":259346,"significance":"If the survey is accurate, it provides a useful map of a scattered literature and a convenient entry point for researchers. The reproduced proofs I checked are logically coherent, and the tables of exact values and asymptotic results are valuable reference material. The paper's value is therefore heavily dependent on the fidelity of its attributions and displayed formulas. I also checked the suspected inconsistency in the proof of Theorem 3: the chain 2+n+floor(n^2/4)=|F|≤1+n+ex(n,K3)=1+n+floor(n^2/4) is a valid contradiction, and the lower-bound construction has size floor(n^2/4)+n+1, exactly one less than the claimed extremal value, which is the intended tightness. No correction to Theorem 3 is needed.","major_comments":[{"comment":"There are clear attribution mismatches that undermine the survey's reference value. The introduction states that 'Hajnal [18] introduced the arrow relation', but reference [18] is Bondy (1972), and Section 4 correctly attributes the relevant result to Bondy. Similarly, Theorem 7 is labeled 'Bollobás [50]' and the introduction credits 'Bollobás [50]' with studying the single-element-removal problem, yet reference [50] is Lovász's book. These are not isolated typos: the authors should verify every name-reference pair in the manuscript.","section":"§1 and §4"},{"comment":"Several displayed formulas have lost their superscripts and are formally wrong as printed. For example, Theorem 21 states 'Tr(n,nr,αn)≥(1−o(1))nλr' instead of 'Tr(n,n^r,αn)≥(1−o(1))n^{λr}', and the preceding paragraph writes 'n α r' where 'n^{α r}' is meant. Theorem 43 in Section 8 similarly has missing exponents, e.g., '(n/(s−1))^{s−1}+o(n^{s−1})' is not what is printed. Because a reader cannot verify the statements in this form, a systematic correction of all mathematical displays is required.","section":"§5, Theorem 21 and surrounding text; also §8, Theorem 43"}],"minor_comments":[{"comment":"The definition of order-shattering appears garbled: the condition 'T′∩C=T′∩D' for all C∈F0 and D∈F1 cannot hold for 2^{|T|} sets unless T′ is empty. Please check the statement against Anstee, Rónyai, and Sali [12] and clarify the quantifiers.","section":"§6, order-shattering definition"},{"comment":"The proof would benefit from explicitly stating that the argument assumes n≥3 and that the hereditary family F contains a 2-set, so that ∅ and all n singletons are indeed members of F; the current text leaves these edge cases implicit.","section":"§3, proof of Theorem 3"},{"comment":"The notation 'F={F⊆[n] : F∈2^{U_i}\\setminus G_i for some i∈[k]}' should make explicit that the empty set is counted once despite lying in every 2^{U_i}; otherwise the displayed size formula is not immediate.","section":"§4, Construction 10"},{"comment":"There are minor typos: 'trival' should be 'trivial' and 'propositon' should be 'proposition'.","section":"§2"},{"comment":"The note that 'all values in this table are precisely established' is too strong: the b=13 row is only for n≥25, and the b=12 row has a separate n=6 exception. These restrictions should be stated in the note.","section":"§3, Table 1"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is a survey with no new results, so its publication value depends on the reliability of its citations and formulas. The mathematical core I checked is sound, but the attribution errors and broken exponents are load-bearing for a reference work of this kind. I recommend requiring a careful pass over all name-reference pairs and all displayed mathematics before acceptance."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper is exactly what it says: a survey of recent advances on arrow relations and traces of sets. No new theorems, but that is not a flaw—it is a useful, well-structured map of a subfield that has moved quickly in the last few years. It covers Frankl's fundamental lemma, defect Sauer results, single-element removal, trace functions, extremal families, forbidden configurations in matrices, Turán numbers for traces, and a list of open problems, including very recent work by Li–Ma–Rong and Reiher–Schülke.\n\nThe reproduced proofs I checked are correct. In particular, the proof of Theorem 3 (Frankl's m(n,3,7)) is logically sound. The stress-test note worried that the displayed inequality 2+n+floor(n^2/4) = |F| ≤ 1+n+floor(n^2/4) is impossible as written; that is exactly the point—it is the contradiction that establishes the upper bound. The lower-bound construction has size floor(n^2/4)+n+1, so the theorem statement is consistent. That flag does not hold up on reading.\n\nThe real soft spots are smaller but real. The introduction credits Hajnal [18] with introducing the arrow relation, but reference [18] is Bondy (1972), and Section 4 correctly attributes the same result to Bondy. That is a genuine citation error in a survey whose value depends on fidelity. There are also several typesetting glitches—Theorem 21's n^r appears as \"nr\", Theorem 43's exponent is mangled, and the table after Theorem 18 could be misread. None of these change the mathematics, but they need cleaning if this is going to function as a reference.\n\nOverall, the survey is accurate and well organized. The misattribution and typos are minor in the sense that they are easy to fix, but they are not trivial in a paper whose whole purpose is orientation and reference. I would send it to peer review rather than desk-reject: referees can verify the attributions and clean up the presentation. The intended audience is newcomers to extremal set theory and trace problems, and for them this could be a genuine entry point. I would not cite it in my own work (I would cite the original papers), but I would assign it to a student looking for a quick overview.","headline":"A well-organized survey of recent arrow-relation results, with a few genuine citation and typesetting errors but no fatal flaws; the stress-test \"inconsistency\" in Theorem 3's proof turns out to be a misreading.","tokens_in":23587,"tokens_out":4610,"would_cite":false,"duration_ms":48217,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05D05","05C35","05C65"],"pacs":[],"model":"deepseek-v4-flash","headline":"This survey assembles recent progress on arrow relations — the extremal framework that quantifies when large set families force rich traces — into a single map of exact bounds, methods, and open problems.","keywords":["arrow relation","traces of sets","extremal set theory","Sauer–Shelah lemma","VC dimension","trace function","hereditary families","Turán numbers for traces"],"falsifier":"Spot-check the survey against its sources: verify that reference [18], credited in the introduction with introducing the arrow relation, actually contains that definition (Section 4's attribution suggests a mismatch), and verify the numerical entries $m(6,4,12) = 24$, $m(11) = 53/10$, and $m(12) = 28/5$ against [17], [49], and [59]; the first confirmed mismatch would directly falsify the survey's accuracy claim.","tokens_in":22505,"feed_emoji":"🔢","tokens_out":20370,"duration_ms":196399,"temperature":0.7,"pith_summary":"This survey assembles the recent literature on arrow relations, the extremal-set-theory framework that asks how large a family $\\mathcal{F} \\subseteq 2^{[n]}$ must be before some $a$-element subset $T$ leaves a trace $\\mathcal{F}|_T$ of at least $b$ distinct sets, written $(n,m) \\to (a,b)$. Its aim is to give researchers a single coherent map of the field: exact values and asymptotic thresholds where they are known, the methods that produce them, and the questions still open. The paper argues that this one relation ties together the Sauer–Shelah lemma, defect-Sauer values $m(n,a,b)$, single-element removal problems, trace functions, shattering-extremal families, forbidden matrix configurations, and Turán-type results for hypergraph traces, and that recent work has filled in precise answers across several fronts. A reader who wants the current state of any of these problems can find the sharpest known bound and its proof strategy in one place, along with a short list of open conjectures.","feed_headline":"Survey pins down when set families must force rich traces","feed_subtitle":"Arrow relations fix how many distinct traces a set family must leave; the best bounds are now in one survey.","key_machinery":"The load-bearing mechanism is Frankl's reduction: Lemma 1 of the paper proves that $(n,m) \\to (a,b)$ holds for all families if and only if it holds for every hereditary family of size exactly $m$. The proof works through the squash operation $S_v(\\mathcal{F})$, which removes a vertex $v$ from a set when the smaller set is not already present; Proposition 2 shows squashing never increases any trace size, so any counterexample with minimum total size must be hereditary. This turns trace problems into minimum-degree and weight problems on hereditary families. Around this core the survey relies on several further devices: the Kruskal–Katona theorem and Katona's weighted version, which drive the single-element-removal lower bounds; the Sparse Kruskal–Katona theorem, which yields the trace-function exponent $\\mu(r,\\alpha)$; the dual inequalities $|str(\\mathcal{F})| \\le |\\mathcal{F}| \\le |tr(\\mathcal{F})|$ and the order-shattering machinery, which characterize $s$-extremal families; and the incidence-matrix and induced-Berge-copy dictionaries that convert trace avoidance into forbidden configurations and Turán-type problems. Each of these tools is stated with enough precision that a reader can apply it directly.","core_discovery":"The central claim is organizational and the paper states it directly: the arrow relation $(n,m) \\to (a,b)$ is a working quantitative language for trace problems, and the results surveyed here show that the language now carries exact answers in several regimes. Concretely, the survey covers: Frankl's reduction ([27], Lemma 1) showing arrow relations need only be verified on hereditary families, via squash operations that never increase any trace; the defect-Sauer values for $a \\le 4$, including $m(n,3,7) = \\lfloor n^2/4 \\rfloor + n + 2$ and $m(n,4,13)$ for $n \\ge 25$, with the three unresolved cells tied to classical Turán problems; the single-element removal function $m(n,s)$ and its limit $m(s)$, now determined for all $s \\le 16$ with $m(11) = 53/10$ and the full range $m(2^{d-1}-c) = (2^d-c)/d$ for $1 \\le c \\le d-1$; trace-function thresholds given by the Sparse Kruskal–Katona theorem, $\\mathrm{Tr}(n,n^r,\\alpha n) = n^{\\mu(1-o(1))}$; the structure and enumeration of families with $|tr(\\mathcal{F})| = |\\mathcal{F}|$; forbidden-configuration bounds in simple matrices; and the reduction of Turán numbers for induced traces, $\\mathrm{ex}(n, \\mathrm{Tr}_r(F)) = \\Theta(\\max_{2 \\le s \\le r} \\mathrm{ex}(n,K_s,F))$. Treated together, these lines of work form a single field with shared tools — squashing, Kruskal–Katona-type weight inequalities, and the trace/shattering duality — and the survey's open problems section names the exact next targets.","pith_inferences":["Because Proposition 2 guarantees that squashing never increases trace size, any exhaustive search for the remaining open cells, such as $m(n,4,9)$, can be restricted to hereditary families; a computational campaign for moderate $n$ would produce data that simultaneously constrains $\\mathrm{ex}(n,\\{C_3^+, C_4\\})$, since Table 1 binds the two quantities together.","The known exact values of $m(s)$ cluster at loss rates $s$ just below powers of two, which suggests a structural transition in the optimal families as $s$ approaches or exceeds a power of two; testing where the extremal constructions change shape would extend the map beyond the boundaries the survey draws.","The dictionaries of Sections 7 and 8 point to a transfer principle that could run in the opposite direction as well: new constructions for hypergraph Turán densities can be converted into trace-avoiding set families through incidence matrices, potentially feeding back into the three open cells of Table 1."],"forward_implications":["The tables of Section 3 settle the minimum family size forcing a trace of size $b$ on $a$ elements for all $a \\le 4$, leaving exactly three cells ($m(n,4,9)$, $m(n,4,14)$, $m(n,4,15)$) whose values are governed by longstanding Turán problems, so progress on those extremal problems would automatically fill the remaining cells.","All values of $m(s)$ for $s \\le 16$ are now known, including $m(11) = 53/10$ confirming a 1994 conjecture of Watanabe and Frankl, and the formula $m(2^{d-1}-c) = (2^d-c)/d$ for $1 \\le c \\le d-1$ completes the picture for loss rates just below powers of two; what remains is $s$ far from powers of two or slightly above them.","The trace function is determined up to logarithmic factors: $\\mathrm{Tr}(n,n^r,\\alpha n) = n^{\\mu(1-o(1))}$ for constant $r, \\alpha$, with $\\mathrm{Tr}(n,n^2,n/2) = \\tilde{\\Theta}(n^{1.7067...})$ closing the long gap between the old polynomial lower bound and $o(n^2)$ upper bound.","The reverse Sauer inequality and the chunk criteria give a structural handle on shattering-extremal families, and the enumeration $f(n,k) = n^{(1+o(1))\\binom{n}{k}}$ shows their number is understood asymptotically; the open Conjecture 4 asks whether every such family can be shrunk by one set while remaining extremal.","Turán numbers for induced traces reduce to generalized Turán numbers via $\\mathrm{ex}(n, \\mathrm{Tr}_r(F)) = \\Theta(\\max_{2\\le s \\le r} \\mathrm{ex}(n, K_s, F))$, so trace-avoidance problems inherit the full extremal-graph toolkit, and the Mubayi–Zhao Conjecture 5 gives a concrete target for the exact values."],"supporting_citations":[{"why":"Supplies the foundational reduction of arrow relations to hereditary families via squashing (Lemma 1, Proposition 2), the defect-Sauer value $m(n,3,7)$, and the single-element-removal lower bound that Sections 2–4 build on.","marker":"[27]"},{"why":"Introduces the defect-Sauer framing and the trace function, proves the lower bound for $\\mathrm{Tr}$, determines $m(n,4,12)$, and establishes the s-extremal equivalences used in Sections 3, 5 and 6.","marker":"[17]"},{"why":"Computes $m(n,4,13)$ for $n \\ge 25$ and the four-vertex trace table, and poses Conjecture 2 on the general $\\ell$-vertex bound.","marker":"[31]"},{"why":"Proves that the limit $m(s)$ exists, determines several early values, and states the conjecture later confirmed by $m(11) = 53/10$.","marker":"[71]"},{"why":"Determines $m(12) = 28/5$ and the first range of $m(2^{d-1}-c)$ values, with constructions that later work extends.","marker":"[59]"},{"why":"Proves $m(11) = 53/10$ and extends the $2^{d-1}-c$ formula for $d \\ge 50$, the two results presented in Section 4.","marker":"[49]"},{"why":"Closes the gap $6 \\le d \\le 49$ and completes the formula $m(2^{d-1}-c) = (2^d-c)/d$ for all $1 \\le c \\le d-1$.","marker":"[63]"},{"why":"Proves the Sparse Kruskal–Katona theorem and derives the trace-function thresholds $\\mathrm{Tr}(n,n^r,\\alpha n) = n^{\\mu(1-o(1))}$ that anchor Section 5.","marker":"[4]"},{"why":"Proves the reverse Sauer inequality $|str(\\mathcal{F})| \\le |\\mathcal{F}|$ and the chunk criteria that characterize families with $|tr(\\mathcal{F})| = |\\mathcal{F}|$ in Section 6.","marker":"[16]"},{"why":"Equates Turán numbers for induced traces with generalized Turán numbers, the main theorem of Section 8.","marker":"[32]"}],"fun_headline_variants":["Arrow relations survey: exact trace bounds in key cases","Trace limits for set families: one survey, sharp answers","Set traces: survey settles many arrow relation bounds","From squashing to shattering: arrow relation trace survey"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The survey's worth rests on the fidelity of its reporting: every theorem, value, and attribution must match the source it cites, and that premise is already strained by an internal slip, since the introduction credits 'Hajnal' with introducing the arrow relation while reference [18] is the 1972 Bondy paper that Section 4 later credits with the same result, so the reader cannot assume every citation was checked against its source.","fun_headline_variants_meta":{"raw":{"variants":["Arrow relations survey: exact trace bounds in key cases","Trace limits for set families: one survey, sharp answers","Set traces: survey settles many arrow relation bounds","From squashing to shattering: arrow relation trace survey"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000512,"raw_usage":{"total_tokens":2555,"prompt_tokens":1077,"completion_tokens":1478,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":693,"completion_tokens_details":{"reasoning_tokens":1414}},"tokens_in":693,"tokens_out":1478,"duration_ms":14660,"temperature":1.0,"reasoning_tokens":1414,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T10:48:59.050418+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Spot-check the survey against its sources: verify that reference [18], credited in the introduction with introducing the arrow relation, actually contains that definition (Section 4's attribution suggests a mismatch), and verify the numerical entries $m(6,4,12) = 24$, $m(11) = 53/10$, and $m(12) = 28/5$ against [17], [49], and [59]; the first confirmed mismatch would directly falsify the survey's accuracy claim.","supporting_citations":[{"cited_title":"Journal of Combinatorial Theory, Series A 34(1), 41–45 (1983)","cited_arxiv_id":null,"evidence_quote":"Supplies the foundational reduction of arrow relations to hereditary families via squashing (Lemma 1, Proposition 2), the defect-Sauer value $m(n,3,7)$, and the single-element-removal lower bound that Sections 2–4 build on."},{"cited_title":"Journal of Combinatorial Theory, Series A 72(2), 189–208 (1995)","cited_arxiv_id":null,"evidence_quote":"Introduces the defect-Sauer framing and the trace function, proves the lower bound for $\\mathrm{Tr}$, determines $m(n,4,12)$, and establishes the s-extremal equivalences used in Sections 3, 5 and 6."},{"cited_title":"Graphs and Combinatorics 40(1) (2024)","cited_arxiv_id":null,"evidence_quote":"Computes $m(n,4,13)$ for $n \\ge 25$ and the four-vertex trace table, and poses Conjecture 2 on the general $\\ell$-vertex bound."},{"cited_title":"Graphs and Combinatorics 10(2-4), 283–292 (1994)","cited_arxiv_id":null,"evidence_quote":"Proves that the limit $m(s)$ exists, determines several early values, and states the conjecture later confirmed by $m(11) = 53/10$."},{"cited_title":"Journal of Combinatorial Theory, Series A 182, 105447 (2021) References 19","cited_arxiv_id":null,"evidence_quote":"Determines $m(12) = 28/5$ and the first range of $m(2^{d-1}-c)$ values, with constructions that later work extends."},{"cited_title":"Exact results on traces of sets","cited_arxiv_id":"2406.18870","evidence_quote":"Proves $m(11) = 53/10$ and extends the $2^{d-1}-c$ formula for $d \\ge 50$, the two results presented in Section 4."},{"cited_title":"Minimum degree in simplicial complexes","cited_arxiv_id":"2501.01294","evidence_quote":"Closes the gap $6 \\le d \\le 49$ and completes the formula $m(2^{d-1}-c) = (2^d-c)/d$ for all $1 \\le c \\le d-1$."},{"cited_title":"Journal of the London Mathematical Society 100(2), 498–517 (2019)","cited_arxiv_id":null,"evidence_quote":"Proves the Sparse Kruskal–Katona theorem and derives the trace-function thresholds $\\mathrm{Tr}(n,n^r,\\alpha n) = n^{\\mu(1-o(1))}$ that anchor Section 5."},{"cited_title":"Proceedings of the London Mathematical Society s3-58(1), 153–168 (1989)","cited_arxiv_id":null,"evidence_quote":"Proves the reverse Sauer inequality $|str(\\mathcal{F})| \\le |\\mathcal{F}|$ and the chunk criteria that characterize families with $|tr(\\mathcal{F})| = |\\mathcal{F}|$ in Section 6."},{"cited_title":"European Journal of Combinatorics 111, 103692 (2023)","cited_arxiv_id":null,"evidence_quote":"Equates Turán numbers for induced traces with generalized Turán numbers, the main theorem of Section 8."}],"review_version":1}