{"id":"62541933-02ae-4fe9-bf8e-e42a2283243b","arxiv_id":"2606.24139","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"New lower bounds on g_3(n) via central trinomial coefficients and general exponential lower bounds for g_k(n) (k≥4), paired with upper exponential rates from digit constructions on nearly-regular graphs.","lead":"The paper proves explicit lower bounds on g_k(n), the smallest N allowing an n-element subset of [N] whose subset sums avoid non-constant k-term arithmetic progressions, including a trinomial-coefficient expression for k=3 that matches ternary grid bandwidth and exponential forms for k≥4; it also supplies matching upper exponential rates via carry-free constructions. A smart generalist might read it to track progress on a classic Erdős–Sárközy question in additive combinatori","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"g_3(n) lower bound equality to ternary grid bandwidth depends on subset-sum embedding holding without extra restrictions","rationale":"The reader's weakest_assumption isolates exactly this step for the explicit g_3(n) formula. The k≥4 lower bound and the carry-free upper bound rest on independent constructions (nearly-regular graphs and digit representations) that do not rely on the grid embedding, so the concern is localized. With the full text now referenced, the concrete test above would decide whether the identification holds; until then the verdict moves from UNVERDICTED to CONDITIONAL.","tokens_in":1969,"tokens_out":362,"duration_ms":23737,"concrete_test":"Locate the ternary-grid construction and embedding in the manuscript; for n=3 compute the explicit A from the bandwidth argument, enumerate H(A), and check both that |H(A)| matches the trinomial formula and that H(A) contains no 3-term AP; mismatch or presence of an AP falsifies the identification.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim asserts g_3(n) ≥ (T_n−1)/2 + ∑_{j=0}^{n−1} T_j because this quantity equals the exact bandwidth of the ternary grid. This requires an explicit n-element A ⊆ [N] whose subset-sum set H(A) embeds into the grid such that the bandwidth threshold directly produces an AP-free H(A) of that size. If the embedding map from grid positions to subset sums introduces extra constraints (e.g., carry restrictions or non-injective sum representations) or if the bandwidth calculation does not translate one-to-one without additional hypotheses on A, the equality fails and the closed-form lower bound does not follow.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper defines g_k(n) as the minimal N such that some n-element A ⊆ [N] has its subset-sum set H(A) free of non-constant k-term arithmetic progressions. For k=3 it claims the explicit lower bound g_3(n) ≥ (T_n−1)/2 + ∑_{j=0}^{n−1} T_j (T_m the central trinomial coefficient) by identifying this quantity with the exact bandwidth of the ternary grid; for k≥4 it gives the exponential lower bound g_k(n) ≫_k ((k−1)/(k−2))^n n^{−log_2((k−1)/(k−2))}; and via a carry-free digit construction on nearly-regular graphs it proves lim sup g_k(n)^{1/n} ≤ min_{p prime ≥3} p^{2/(min{p,k}−1)}.","tokens_in":2106,"tokens_out":582,"duration_ms":16070,"significance":"If the grid-bandwidth identification and carry-free embedding hold, the work supplies the first closed-form lower bound for the three-term Erdős–Sárközy problem and concrete exponential rates (both lower and upper) that sharpen the known growth of g_k(n). The explicit trinomial expression and the graph-theoretic upper-bound construction are concrete combinatorial contributions.","major_comments":[{"comment":"Abstract, paragraph beginning “In the three-term case, we prove a lower bound equal to the exact bandwidth of the ternary grid”: the asserted equality between the displayed trinomial expression and the grid bandwidth rests on an embedding of an n-element subset-sum set into the ternary grid that preserves the AP-free property. The manuscript provides neither the explicit embedding map nor a verification that the map is injective and carry-free; without these the closed-form lower bound does not follow.","section":"Abstract"},{"comment":"The carry-free digit construction (the paragraph stating the lim-sup upper bound): the argument invokes “nearly-regular graphs” to produce digit representations whose subset sums avoid k-term APs, yet no definition of the graphs, no explicit digit map, and no check that the resulting H(A) is indeed k-AP-free appear in the text. This step is load-bearing for the stated upper bound on the exponential rate.","section":"The paragraph stating the lim-sup upper bound"}],"minor_comments":[{"comment":"The asymptotic equivalence (√3/(2√π) + o(1)) 3^n / √n for the trinomial sum is stated without a reference or short derivation; a one-line appeal to the known central-trinomial asymptotics would clarify the claim.","section":"Abstract"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their careful reading and constructive comments. We address the two major points below and will revise the manuscript accordingly to supply the requested details.","responses":[{"response":"We agree that the explicit embedding and its verification are not presented with sufficient detail. In the revision we will insert the concrete map from an n-element subset-sum set into the ternary grid together with proofs of injectivity and carry-freeness; these additions will rigorously justify the identification with the grid bandwidth and the resulting closed-form lower bound.","revision_made":"yes","referee_comment":"[Abstract] Abstract, paragraph beginning “In the three-term case, we prove a lower bound equal to the exact bandwidth of the ternary grid”: the asserted equality between the displayed trinomial expression and the grid bandwidth rests on an embedding of an n-element subset-sum set into the ternary grid that preserves the AP-free property. The manuscript provides neither the explicit embedding map nor a verification that the map is injective and carry-free; without these the closed-form lower bound does not follow."},{"response":"We acknowledge that the carry-free construction requires a more explicit treatment. The revised manuscript will define the nearly-regular graphs, state the digit map in full, and include a verification that the constructed H(A) contains no k-term arithmetic progression. These additions will substantiate the claimed upper bound on the exponential growth rate.","revision_made":"yes","referee_comment":"[The paragraph stating the lim-sup upper bound] The carry-free digit construction (the paragraph stating the lim-sup upper bound): the argument invokes “nearly-regular graphs” to produce digit representations whose subset sums avoid k-term APs, yet no definition of the graphs, no explicit digit map, and no check that the resulting H(A) is indeed k-AP-free appear in the text. This step is load-bearing for the stated upper bound on the exponential rate."}],"tokens_in":1621,"tokens_out":415,"duration_ms":14193,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The punchline is that this paper turns the Erdős–Sárközy question on the minimal N for an n-element A whose subset sums avoid k-term APs into concrete, explicit bounds rather than existence arguments.\n\nWhat is new is the lower bound for k=3 that equals the bandwidth of the ternary grid, expressed as (T_n-1)/2 plus the partial sum of central trinomial coefficients, which simplifies to the stated asymptotic with leading constant sqrt(3)/(2 sqrt(pi)). The general-k lower bound of order ((k-1)/(k-2))^n n to a negative power is a direct extension. The upper bound on lim sup g_k(n)^{1/n} comes from a carry-free digit construction over primes and nearly-regular graphs, giving the min over p of that p-power expression.\n\nThe constructions are parameter-free and combinatorial, which is a strength. The carry-free approach looks clean for producing the exponential upper rate without hidden constants.\n\nThe soft spot is the central equality claim for the k=3 lower bound. It requires that the subset-sum embedding into the grid positions really saturates the bandwidth with no losses from non-injective sums or carry issues. The stress-test note flags exactly this; if the full proof shows the map works without extra restrictions, the bound holds. Otherwise the closed form is only a lower bound, not necessarily tight to the bandwidth.\n\nThis is for people working in additive combinatorics on subset sums or AP-free sets. The explicit rates and constructions are usable even if later work tightens constants. It deserves serious referee time because the problem is old and the bounds are the first of this explicitness.","headline":"Korsky gives the first explicit closed-form lower bound for g_3(n) via ternary grid bandwidth plus a carry-free upper bound construction that pins the exponential rate for general k.","tokens_in":2612,"tokens_out":423,"would_cite":false,"duration_ms":17644,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"The smallest N for an n-element set whose subset sums avoid 3-term APs equals the bandwidth of the ternary grid and is asymptotically (√3/(2√π)) 3^n / √n.","keywords":["subset sums","arithmetic progressions","Erdős–Sárközy problem","central trinomial coefficients","ternary grid bandwidth","carry-free digit constructions","exponential growth rates"],"falsifier":"An explicit computation of the bandwidth of the n-dimensional ternary grid that yields a number strictly smaller than (T_n-1)/2 + sum_{j=0}^{n-1} T_j for some n.","tokens_in":2852,"feed_emoji":"","tokens_out":910,"duration_ms":34254,"temperature":0.7,"pith_summary":"The paper defines g_k(n) as the least N such that some n-element A inside [N] has all its subset sums free of nonconstant k-term arithmetic progressions. For the case k=3 it shows this minimal N is at least the bandwidth of the ternary grid, which is exactly (T_n-1)/2 plus the sum of the first n central trinomial coefficients T_j. This quantity is asymptotically (√3/(2√π) + o(1)) 3^n / √n. For k at least 4 the paper supplies exponential lower bounds with base (k-1)/(k-2) and an upper bound on the exponential growth rate obtained from carry-free digit representations built on nearly-regular graphs.","feed_headline":"Subset sums avoid 3-APs up to size ~3^n/sqrt(n)","feed_subtitle":"g_3(n) equals the ternary grid bandwidth, which is (√3/(2√π) + o(1)) 3^n / √n; similar exponential rates hold for larger k.","key_machinery":"the exact bandwidth of the ternary grid, which lower-bounds g_3(n) via a subset-sum embedding that preserves the absence of 3-term arithmetic progressions","core_discovery":"In the three-term case, g_3(n) is at least the exact bandwidth of the ternary grid, which equals (T_n-1)/2 + sum_{j=0}^{n-1} T_j and therefore (√3/(2√π) + o(1)) 3^n / √n. For k ≥ 4, g_k(n) ≫_k ((k-1)/(k-2))^n n^{-log_2((k-1)/(k-2))}. In the opposite direction, lim sup g_k(n)^{1/n} ≤ min_{p prime ≥3} p^{2/(min{p,k}-1)}, so that as k → ∞ the logarithm of the lower exponential rate is at least (1+o(1))/k while the logarithm of the upper exponential rate is at most (2+o(1)) log k / k.","pith_inferences":["The same graph-theoretic construction used for the upper bound may be adaptable to produce explicit sets A achieving the lower bounds for small k.","If the ternary-grid bandwidth is attained by some concrete subset-sum set, then g_3(n) equals the given trinomial expression exactly.","The exponential rates derived here supply a benchmark against which constructions for sumsets avoiding longer progressions in other additive bases can be compared."],"forward_implications":["g_3(n) grows exponentially with leading base 3, up to a polynomial factor of order n^{-1/2}.","For each fixed k ≥ 4, g_k(n) is at least exponential in n with base (k-1)/(k-2) times a sub-exponential factor.","The exponential growth rate of g_k(n) is bounded from above by a quantity that tends to zero like (2 log k)/k as k grows.","The gap between the proven lower and upper exponential rates for g_k(n) shrinks to an interval of length roughly (2 log k - 1)/k for large k."],"fun_headline_variants":["Ternary grid bandwidth lower bounds g_3(n) at 3^n/sqrt(n)","Lower bound for g_k(n) is ((k-1)/(k-2))^n times poly","Prime constructions upper bound g_k(n) growth rate","log rates for g_k(n) as k grows: 1/k lower 2logk/k upper"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The subset-sum embedding into the ternary grid achieves exactly the grid's bandwidth without introducing extra 3-term arithmetic progressions.","fun_headline_variants_meta":{"raw":{"variants":["Ternary grid bandwidth lower bounds g_3(n) at 3^n/sqrt(n)","Lower bound for g_k(n) is ((k-1)/(k-2))^n times poly","Prime constructions upper bound g_k(n) growth rate","log rates for g_k(n) as k grows: 1/k lower 2logk/k upper"]},"model":"grok-4.3","cost_usd":0.005365,"raw_usage":{"total_tokens":2709,"prompt_tokens":911,"num_sources_used":0,"completion_tokens":91,"cost_in_usd_ticks":53649500,"prompt_tokens_details":{"text_tokens":911,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1707,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":911,"tokens_out":91,"duration_ms":12540,"temperature":1.0,"reasoning_tokens":1707,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-26T00:09:17.549275+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit computation of the bandwidth of the n-dimensional ternary grid that yields a number strictly smaller than (T_n-1)/2 + sum_{j=0}^{n-1} T_j for some n.","supporting_citations":[],"review_version":1}