{"id":"3444c245-f6be-451c-beb1-a4d058aa2702","arxiv_id":"2607.17746","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For k≥5, infinitely many n have exactly 2^{r_k(n)(1+o(1))} k-AP-free subsets of [n]; for all n and k≥3 the count is 2^{O(r_k(n))}.","lead":"This paper counts the number of subsets of {1,...,n} that contain no k-term arithmetic progression. For k≥5, it proves that for infinitely many n this number is exactly 2^{r_k(n)(1+o(1))}, where r_k(n) is the size of the largest such subset — settling Cameron–Erdős for infinitely many scales.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Appendix A degree miscalculation breaks Rankin-type lower bounds for multidimensional and linear-equation applications","rationale":"The main Theorem 1.3 for k-APs appears logically sound: it relies on the known Rankin bound (2) and the smoothness argument in Lemma 4.2 works for α<1/2. The reader's weakest assumption (the smoothness condition) is a true limitation but not a flaw. However, the paper's advertised applications to multidimensional Szemerédi and to almost all linear systems depend on Propositions 6.2 and 6.3, whose proofs contain a clear degree error in the base case of the Rankin-type construction. This is an internal inconsistency, not a matter of disagreement with consensus. The verdict remains CONDITIONAL because the main k-AP theorem may still be correct and the appendix may be repairable, but the authors must fix the lower-bound proofs before the broader framework is accepted.","tokens_in":30708,"tokens_out":43985,"duration_ms":422183,"concrete_test":"Analytically check the degree: with ℓ=2 and P_i of degree 3, ΣP_i^2 has degree 6; a nonzero degree-6 polynomial can have 5 distinct real roots, so the root-counting step fails. Then attempt to construct (e.g., numerically) a cubic curve in R^3 intersecting a sphere in 5 points; if such points exist, the base-case claim that the sphere is (X,3)-free for |X|=5 is false, confirming the concern.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Appendix A, Lemma A.4 claims the integer sphere B_r is (X, 2ℓ−1)-free. The proof takes P_1,...,P_t of degree at most 2ℓ−1 and considers Q = ΣP_i^2 − R. It asserts Q has degree at most 2ℓ and hence, with 2ℓ+1 roots, is zero. But deg(P_i^2) ≤ 4ℓ−2, so deg(Q) ≤ 4ℓ−2, which exceeds 2ℓ for ℓ ≥ 2. A nonzero real polynomial of degree 4ℓ−2 can vanish at 2ℓ+1 points; the contradiction does not follow. The same misstep appears in Lemma A.11 for parametric representations, and it invalidates Propositions 6.2 and 6.3 — the Rankin-type lower bounds needed to apply Theorem 2.6 to multidimensional X-free sets (Theorem 1.8) and to almost all linear systems (Theorem 1.5). Theorem 1.3 for k-APs is insulated because it cites Rankin's bound (2), but the advertised broader framework currently rests on an invalid proof.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a general hypergraph-container framework for counting subsets of [n]^d that avoid solutions to translation-invariant systems of linear equations. The two abstract theorems, Theorem 2.6 and Theorem 2.7, assert that if the extremal function r_A(n) satisfies a Behrend-type lower bound n^d exp(-O(log^α n)) with α<1/2 or α<1 respectively, then the number of A-free sets is, respectively, 2^{(1+o(1))r_A(n)} for infinitely many n or 2^{O(r_A(n))} for all n. From these the paper derives: for k≥5, the number of k-AP-free subsets of [n] is 2^{r_k(n)(1+o(1))} for infinitely many n (Theorem 1.3), and for k≥3, it is 2^{O(r_k(n))} for all n (Theorem 1.4); analogous multidimensional and almost-all linear equation results (Theorems 1.5, 1.6, 1.8, 1.9). The proofs use a new supersaturation statement phrased as absence of cheap covers, a smoothness lemma passing to a subsequence, and an iterative container-shrinking argument.","tokens_in":30914,"tokens_out":14593,"duration_ms":135348,"significance":"If the main framework is valid, Theorem 1.3 is a substantial advance on the Balogh–Liu–Sharifzadeh result and resolves the Cameron–Erdős question in the qualitative form for infinitely many n. Theorem 1.4 is also notable and is best possible up to a constant factor in the exponent. The abstract container formulation, with supersaturation statements in the cheap-cover language, is elegant and likely to be reusable. The paper is careful in transferring the extremal condition to the counting exponent, and the proofs of Propositions 4.1 and 5.3 are largely transparent. However, the advertised generality depends on an appendix whose degree estimates are wrong in a load-bearing way, and the proof of Theorem 5.4 has a constant-factor gap. These issues do not affect the k-AP results, but they must be repaired before the broader claims (Theorems 1.5 and 1.8) can be accepted.","major_comments":[{"comment":"The proof of Lemma A.4 asserts that Q = ΣP_i^2 − R has degree at most 2ℓ. Since each P_i has degree at most 2ℓ−1, deg(Q) ≤ 4ℓ−2, not 2ℓ. Thus a nonzero Q can vanish at 2ℓ+1 points, and the conclusion that the polynomial copy is trivial does not follow for ℓ≥2. The same miscalculation appears in Lemma A.11. This is load-bearing: Propositions 6.2 and 6.3, and therefore Theorems 1.8 and 1.5, rest on these lemmas. Additionally, Lemma A.4’s statement promises an (X,2ℓ′)-free set, but the application needs an (X,1)-free set; for ℓ=1, ℓ′=0 gives only (X,0)-freeness, which is vacuous. The degree indices are inconsistent throughout the appendix. The argument may be repairable by constructing the sphere as (X,ℓ)-free and then iterating down to degree 1, but as printed the proof is invalid.","section":"Appendix A, Lemmas A.4 and A.11"},{"comment":"In the second case of the iterative step, Proposition 5.3 is applied to a container C with |C| ≥ λ_{i+1}r_A(n). However, the container lemma supplies a cover G of H[C] with w_p(G) ≤ p|C|, not w_p(G) ≤ p|C'| for a subset C' of size exactly λ_{i+1}r_A(n). The inherited weight bound is therefore p|C|, which can be a constant factor larger than the q|B| required by Proposition 5.3. The argument is likely salvageable by choosing p = c λ^{-3} with a sufficiently small constant c to absorb the factor, but this adjustment is not written and the present application has a mismatch of hypotheses.","section":"§5, Theorem 5.4"}],"minor_comments":[{"comment":"The tiling sentence in the border case is garbled: 'with 1 ≤ h_i ≤ t box [t]^d' should explain that each Q_i is a translate of [h_i]^d for some h_i ≤ t, and that |Z'∩Q_i| ≤ r_A(h_i) is then bounded using monotonicity and Lemma 3.1.","section":"§4, Proposition 4.1"},{"comment":"In Appendix A the text attributes the treatment to 'Lacey and Laba', while reference [19] is listed as 'Laba and Lacey'. Please make the citation consistent.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The main k-AP results are likely correct, but the appendix error is serious. If the authors can correct the Rankin-type argument, e.g. by using degree ℓ in the sphere construction and adjusting the induction, the paper would be acceptable. As is, the paper should not be published with Theorems 1.5 and 1.8 resting on the current appendix. The constant-factor mismatch in Theorem 5.4 should also be fixed explicitly."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Let me give you the short version: the k-AP results are the real thing, but the appendix proof that supports the multidimensional and linear-system applications has a degree bound error that looks fatal for those parts as written.\n\nThe paper's main theorem — that for k≥5 the number of k-AP-free subsets of [n] is 2^{r_k(n)(1+o(1))} on an infinite sequence — is a genuine advance. The all-n bound 2^{O(r_k(n))} for k≥3 is also new and matches the conjectured shape. The abstract container/supersaturation framework is well designed; Sections 4 and 5 are carefully argued and, as far as I can tell, internally consistent. The authors also honestly state the k=3,4 obstruction in Section 7. For the k-AP results, they use Rankin's bound directly, so those are insulated from the appendix problems.\n\nThe soft spot is Appendix A. In Lemma A.4 they take polynomials P_i of degree at most 2ℓ−1 and argue that the sum of squares P_1^2+...+P_t^2−R has degree at most 2ℓ, so with 2ℓ+1 roots it must be identically zero. But the degree is actually at most 4ℓ−2. A nonzero polynomial of that degree can vanish at 2ℓ+1 points, so the contradiction does not follow. The same error appears in Lemma A.11 for parametric representations. That means Propositions 6.2 and 6.3 — the Rankin-type lower bounds for multidimensional X-free sets and for almost all linear systems — are not proved by this manuscript. Theorems 1.5, 1.6, 1.8, and 1.9 therefore rest on unsupported claims. The abstract theorems 2.6 and 2.7 may still be fine; it's their advertised applicability that is overreaching.\n\nI also checked the reader's concern about Theorem 5.4. I think that one is a non-issue: Proposition 5.3 is applied to B = f(T), and Theorem 3.6 gives w_p(G) ≤ p|f(T)|, exactly as needed. The λ notation is confusing but not a gap.\n\nBottom line: this is a significant paper for k-AP counting, but the appendix needs a real fix or the applications need to be downgraded to conditional on known Rankin-type bounds. I'd send it to a serious referee, with the expectation of major revision.","headline":"Strong sharp-counting theorem for k-APs, but the appendix proof that powers the broader applications has a degree error that looks fatal for those parts.","tokens_in":31493,"tokens_out":4917,"would_cite":true,"duration_ms":43986,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05D05","11B25","05D40"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that for every k≥5, the number of subsets of {1,...,n} containing no k-term arithmetic progression is 2^{r_k(n)(1+o(1))} for infinitely many n, resolving a long-standing question in extremal combinatorics.","keywords":["arithmetic progressions","k-term progression-free sets","extremal combinatorics","hypergraph containers","supersaturation","translation-invariant systems of equations","multidimensional Szemerédi theorem","counting subsets"],"falsifier":"Find a translation-invariant matrix A and an infinite sequence n_i with r_A(n_i) ≥ n_i^d exp(-O(log^{1/3} n_i)) but for which the number of A-free sets in [n_i]^d is at least 2^{(1+c) r_A(n_i)} for a fixed c>0; this would contradict Theorem 2.6 and its progression corollary. A more targeted probe is to compute the density ratio r_A(n)/n^d versus r_A(n e^{-√log n})/(n e^{-√log n})^d along candidate sequences: if it fails to stay above 1 - δ/8 for every infinite sequence, the proof's smoothness condition is genuinely necessary, and if it ever does so for a sequence where the count is not sharp,","tokens_in":30551,"feed_emoji":"🔢","tokens_out":12919,"duration_ms":111620,"temperature":0.7,"pith_summary":"This paper answers a long-standing counting question: how many subsets of {1,...,n} contain no k-term arithmetic progression? The authors prove that for every k≥5, along an infinite sequence of n, the number is 2^{r_k(n)(1+o(1))}, where r_k(n) is the size of the largest such subset; this matches the trivial lower bound. They also prove that for every k≥3 and every n, the number is at most 2^{O(r_k(n))}. Both results are instances of a general framework for translation-invariant systems of arithmetic equations: if the extremal threshold is at least n^d exp(-O(log^α n)) with α<1/2, the sharp exponent follows; with α<1, the all-n bound follows. The framework also yields analogous counting theorems for multidimensional patterns and for almost all systems of linear equations.","feed_headline":"Count of sets avoiding 5-term progressions is now sharp","feed_subtitle":"For k≥5, the number of subsets with no k-term arithmetic progression equals 2^{r_k(n)(1+o(1))} for infinitely many n.","key_machinery":"The key mechanism is a pair of supersaturation lemmas (Propositions 4.1 and 5.3) that rule out 'cheap covers' of the hypergraph of forbidden configurations. Proposition 4.1 samples a random grid with prime common difference; if the extremal density changes slowly across scales n and n e^{-√log n}, any set of size (1+δ)r_A(n) contains a forbidden configuration outside every low-weight cover. Lemma 4.2 converts the lower bound r_A(n) ≥ n^d exp(-O(log^α n)) with α<1/2 into this smoothness condition for an infinite sequence. For the all-n bound, a supermultiplicativity property of r_A(n) replaces smoothness, and an iterative container-shrinking argument produces containers of size O(r_A(n)). Tog","core_discovery":"The central discovery is that, for any translation-invariant matrix A defining forbidden arithmetic configurations, the number of A-free subsets of [n]^d is controlled by the extremal threshold r_A(n): for an infinite sequence of n it is 2^{(1+o(1))r_A(n)} whenever r_A(n)≥n^d exp(-O(log^α n)) with α<1/2, and for all n it is 2^{O(r_A(n))} whenever α<1. For k-term progressions with k≥5, known lower-bound constructions provide the required α<1/2, so the sharp exponent holds; the best constructions for k=3,4 only reach α=1/2, which the proof's smoothness condition cannot handle. The proof's engine is a supersaturation statement: any set exceeding the extremal threshold by a factor (1+δ) must con","pith_inferences":["If future work improves the lower bounds for 3- and 4-term progressions to the required α<1/2, the same framework would likely settle the counting question for all k and all n.","The unconditional all-n bound suggests that for generic translation-invariant patterns, avoiding sets are spread out enough for the extremal size alone to control the count; deviations from this, as in some classical examples, must stem from special additive structure.","The random-grid supersaturation technique may be adaptable to other hypergraphs defined by polynomial or algebraic configurations, potentially giving new counting theorems outside the linear-equation setting.","The iterative container-shrinking scheme appears to be a reusable tool for extremal counting problems whose extremal function satisfies a supermultiplicativity inequality."],"forward_implications":["For every k≥5, the number of k-term-progression-free subsets of [n] is 2^{r_k(n)(1+o(1))} for infinitely many n, answering the long-standing counting question in the affirmative along that sequence.","For every k≥3 and all n, the number of k-term-progression-free subsets is 2^{O(r_k(n))}, so the trivial lower bound is always tight up to a constant in the exponent.","For any finite pattern X⊂Z^d with |X|≥5, the number of subsets of [n]^d avoiding positive copies of X is 2^{r_X(n)(1+o(1))} for infinitely many n; with |X|≥3, a 2^{O(r_X(n))} bound holds for all n.","For almost all (k,h)-systems of translation-invariant linear equations with enough equations, the number of solution-free sets satisfies the sharp exponent for infinitely many n and a 2^{O(r_A(n))} bound for all n.","The two general theorems unify many previous counting results and show that a sufficiently strong lower bound on the extremal threshold essentially determines the counting exponent."],"fun_headline_variants":["k-AP-free set count matches extremal bound for k≥5","Sharp exponent for counting sets without k-term APs","Cameron-Erdős question answered for infinitely many n","Counting AP-free sets: new general framework","AP-free set counting: sharp for k≥5 infinitely often"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The sharp-exponent theorem rests on the extremal threshold being at least n^d exp(-O(log^α n)) with α<1/2, the condition that guarantees the density smoothness across scales; for 3- and 4-term progressions even the strongest conjectured lower bounds provide only α=1/2, so the main conclusion currently stops at k≥5.","fun_headline_variants_meta":{"raw":{"variants":["k-AP-free set count matches extremal bound for k≥5","Sharp exponent for counting sets without k-term APs","Cameron-Erdős question answered for infinitely many n","Counting AP-free sets: new general framework","AP-free set counting: sharp for k≥5 infinitely often"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000281,"raw_usage":{"total_tokens":1559,"prompt_tokens":860,"completion_tokens":699,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":604,"completion_tokens_details":{"reasoning_tokens":630}},"tokens_in":604,"tokens_out":699,"duration_ms":7467,"temperature":1.0,"reasoning_tokens":630,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T17:05:34.164329+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find a translation-invariant matrix A and an infinite sequence n_i with r_A(n_i) ≥ n_i^d exp(-O(log^{1/3} n_i)) but for which the number of A-free sets in [n_i]^d is at least 2^{(1+c) r_A(n_i)} for a fixed c>0; this would contradict Theorem 2.6 and its progression corollary. A more targeted probe is to compute the density ratio r_A(n)/n^d versus r_A(n e^{-√log n})/(n e^{-√log n})^d along candidate sequences: if it fails to stay above 1 - δ/8 for every infinite sequence, the proof's smoothness condition is genuinely necessary, and if it ever does so for a sequence where the count is not sharp,","supporting_citations":[],"review_version":1}