{"id":"f87bc88e-63ca-4f69-b1e7-65b2d67fe5a7","arxiv_id":"2607.10111","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":5,"one_line_summary":"For every fixed s ≥ 4, f^{(4)}_{s,s+1}(n) = (log n)^{o(1)}, from a new 3-uniform bound f^{(3)}_{s,s+1}(n) = O(log n/log log n).","lead":"Small improvements in the 3-uniform case yield, via a stepping-up argument, the first subpower-of-log-n upper bounds for the 4-uniform Erdős–Rogers function: f^{(4)}_{s,s+1}(n) = (log n)^{o(1)}. This resolves a problem of Conlon, Fox and Sudakov and improves the known upper bounds for all higher uniformities.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the container-based proof of Theorem 1.5 checks out; the only open condition concerns external Lemma 4.4 for Corollary 1.6.","rationale":"The reader's weakest_assumption was the container applicability calculation in Section 3.4, Step 1. I checked this explicitly: the algebra is sound, the exponent margin is comfortably negative (≤ -4/5), and the recursive container count follows. The paper's central result, Theorem 1.5, is self-contained and does not depend on the unproved external Lemma 4.4; that lemma concerns only Corollary 1.6. Thus the reader's conditional verdict is appropriate for the paper as a whole, but the central claim itself should not be downgraded. I therefore leave the verdict unchanged.","tokens_in":17086,"tokens_out":31514,"duration_ms":296840,"concrete_test":"Recompute the Step 1 co-degree ratio using the exact bounds d(H)≥hζt^s/binom(t,2)q and Δℓ(H)≤C_s t^{s-3}q^{h-ℓ} for s=4 (h=5, a=0.02) and s=5; verify that max_{2≤ℓ≤h} [a(h+ℓ-1)-1] ≤ -4/5 for all t large. Independently, locate and verify Lemma 4.4 in [19] to clear the Corollary 1.6 condition.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I find no load-bearing flaw in the central claim. The delicate container applicability in Section 3.4, Step 1 is correct: for H=Γ[C] with e(H)≥ζt^s, the average-degree lower bound gives d(H)≥c t^{s-2}/q, and the co-degree ratio satisfies Δℓ(H)/(d(H)τ^{ℓ-1}) ≤ C t^{a(h+ℓ-1)-1} ≤ C t^{-(8h+1)/(10h)} = O(t^{-4/5}), so Lemma 2.3 applies with τ=t^{-2a}. The recursive container count yields |C|=q^{o(m)}, Claim 3.6 and the product estimate give the q^{-ηm} saving, and the palette Lemma 3.1 and the monotone stepping-up Lemma 4.3 are internally consistent. The only actual condition in the paper is Corollary 1.6, which invokes Lemma 4.4 from [19] without proof; this does not affect the resolution of Problem 1.1.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the hypergraph Erdős–Rogers function f^{(k)}_{s,t}. The main results are: (i) Theorem 1.3, f^{(3)}_{s,s+1}(n) ≤ C_s log n / log log n for every fixed s ≥ 3; (ii) Theorem 1.5, f^{(4)}_{s,s+1}(n) ≤ exp(C_s log log n / log log log n) for every fixed s ≥ 4, which resolves Problem 1.1 of Conlon–Fox–Sudakov; and (iii) Corollary 1.6, f^{(k)}_{k+1,k+2}(n) ≤ exp(C_k log_{(k-2)} n / log_{(k-1)} n) for every fixed k ≥ 5. The 3-uniform proof constructs a random pair-coloring using a K_s-free palette graph F, encodes K_s-free local colorings as independent sets in an auxiliary h-graph Γ, and counts them via recursive hypergraph containers. The 4-uniform result is obtained from the 3-uniform one by a monotone stepping-up construction over binary sequences using δ-chains.","tokens_in":17301,"tokens_out":26633,"duration_ms":249221,"significance":"If correct, Theorem 1.5 is a substantial advance: it gives the first general subpower-in-log n upper bound for f^{(4)}_{s,s+1}(n), resolving a problem stated by Conlon, Fox and Sudakov. Theorem 1.3 also improves the logarithmic Dudek–Mubayi bound by a log log n factor. I checked the load-bearing steps in detail: the probabilistic construction of the palette graph in Lemma 3.1, the co-degree bound in Lemma 3.4, the container applicability calculation in Section 3.4 Step 1 (the exponent a(h+ℓ−1)−1 ≤ −(8h+1)/(10h) < −4/5 gives the required margin), the recursive container count |C| = q^{o(m)}, and the δ-chain/Erdős–Szekeres argument in Lemma 4.3. These all cohere, and the proof of the main theorem is essentially self-contained apart from standard external tools. The main caveat is Corollary 1.6, which depends on Lemma 4.4 imported from [19]; this does not affect the resolution of Problem 1.1.","major_comments":[],"minor_comments":[{"comment":"The bound e(Γ)/(ζt^s) ≤ q^h/ζ drops a factor of 1/s! from (t choose s)/t^s. The resulting O(log q) number of refinement steps is unaffected, but the displayed inequality should be corrected or qualified with an absolute constant.","section":"Section 3.4, Step 2"},{"comment":"The sentence 'iterating the second inequality in Lemma 4.4 k−5 times and then applying the first inequality once' states the order of application backwards: the first (4→5) inequality must be used before iterating the second. The displayed final bound is correct, but the sentence should be rephrased. Also, the monotonicity step used to pass from 2^{n_i} to n_{i−1} is implicit and should be stated explicitly.","section":"Section 4.3, proof of Corollary 1.6"},{"comment":"The induction uses a monotonicity property of binomial coefficients (that binom(x+y, x) increases under the componentwise bounds x ≤ I(A)−1, y ≤ D(A), and x ≤ I(A), y ≤ D(A)−1) that is not stated. The step is valid, but a one-sentence justification would improve readability.","section":"Lemma 4.2"},{"comment":"Corollary 1.6 is entirely contingent on Lemma 4.4, which is quoted from [19] without proof. If [19] is not yet published, the authors should either provide a proof or clearly flag the dependence. This does not affect Theorems 1.3 and 1.5.","section":"Section 4.3, Lemma 4.4"},{"comment":"There are minor typographical and formatting issues, including inconsistent superscript notation for hypergraph cliques (e.g., K_s^3 vs K_s^{(3)}) and some OCR artefacts in the header. These should be cleaned up.","section":"Throughout"}],"recommendation":"minor_revision","confidential_remarks":"The central result is sound and well within the scope of the journal. The only substantive concern is that Corollary 1.6 relies on Lemma 4.4 from [19], an arXiv preprint that shares an author with the present paper. I recommend that the editor either verify that lemma with the referees of [19] or ask the authors to include a proof or appendix. This does not affect the resolution of Problem 1.1."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: this paper does what it says. It proves f^{(4)}_{s,s+1}(n) = (log n)^{o(1)} for fixed s >= 4, settling Problem 1.1 of Conlon, Fox, and Sudakov, and the engine is a new 3-uniform result f^{(3)}_{s,s+1}(n) = O(log n / log log n), which breaks the logarithmic barrier left by Dudek and Mubayi. The higher-uniform corollary is also new, though it carries the one condition mentioned below.\n\nWhat is genuinely new: the 4-uniform subpower bound itself, and the 3-uniform log/log-log improvement. The method is a direct extension of the pair-coloring program of CFS and Dudek-Mubayi, but the growing auxiliary palette combined with recursive hypergraph containers is a real step forward, not a rhetorical twist. I checked the key estimates with some care. The union bound in Lemma 3.1 is sound: the transversal clique probability dominates the number of choices. In Lemma 3.5, Step 1, the container condition works: for H=Γ[C] with e(H)≥ζ t^s, average degree is Ω(t^{s-2}/q), and the co-degree ratio is O(t^{a(h+ℓ-1)-1}) = O(t^{-4/5}), so Lemma 2.3 applies with τ = t^{-2a}. The product estimate then gives the q^{-ηm} saving, and the δ-chain argument in Lemma 4.3 is consistent. I didn't find a load-bearing error.\n\nSoft spots: Corollary 1.6 depends on Lemma 4.4, a stepping-up recursion cited from [19], which is an overlapping-author preprint (Lin is a coauthor) and is not proved in this paper. This doesn't affect Theorems 1.3 and 1.5, which are self-contained modulo standard tools, so it's a minor issue for the main claims, but if you plan to cite the corollary you should check Lemma 4.4 in [19]. The paper is honest in Section 5 that the stepping-up argument loses an exponential factor, which is why Problem 5.1 (polynomial in log log n) remains open; that limitation is stated plainly.\n\nWho this is for: anyone working on generalized Ramsey numbers or hypergraph Erdős–Rogers functions. It resolves an explicit open problem and the container-plus-palette technique is likely to be reusable. It deserves a serious referee; I'd send it out without hesitation, with the request that the referee verify Lemma 4.4 or ask for a proof in a revision.","headline":"Resolves the Conlon–Fox–Sudakov 4-uniform problem with a subpower bound via a new 3-uniform log/log-log estimate; the main proofs are sound and the only caveat is an external lemma for the higher-uniform corollary.","tokens_in":17904,"tokens_out":3357,"would_cite":true,"duration_ms":33792,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C65","05C55","05D40"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that for every fixed s≥4, the 4-uniform Erdős–Rogers function f^{(4)}_{s,s+1}(n) is (log n)^{o(1)}, resolving a problem from the literature.","keywords":["Erdős–Rogers function","hypergraph Ramsey theory","hypergraph containers","stepping-up construction","clique-free induced subgraphs","probabilistic method","3-uniform hypergraphs","4-uniform hypergraphs"],"falsifier":"Enumerate, for a small concrete case such as s=3 and t around a few hundred, the number of local pair-colorings that produce a K_3^{(3)}-free 3-graph. If that count is not at most q^{(1−η)m} with η>0, or if the auxiliary palette F cannot be built with the claimed robust transversal property at q = t^a, then Lemma 3.5 fails and the main theorem collapses.","tokens_in":16895,"feed_emoji":"🧮","tokens_out":6225,"duration_ms":60325,"temperature":0.7,"pith_summary":"The Erdős–Rogers function f^{(k)}_{s,t}(n) measures the largest guaranteed size of a vertex set that spans no K_s^{(k)} inside any n-vertex K_t^{(k)}-free k-uniform hypergraph. The paper proves that for every fixed s≥4, f^{(4)}_{s,s+1}(n) ≤ exp(O(log log n / log log log n)), which is (log n)^{o(1)}; this settles the 4-uniform problem posed in the earlier literature. The engine is a new 3-uniform estimate: f^{(3)}_{s,s+1}(n) ≤ C_s log n / log log n, improving the previous logarithmic upper bound by a factor of log log n. The proof builds a random 3-graph by coloring pairs with a robust clique-free palette, encodes local bad colorings as independent sets of an auxiliary hypergraph, and counts them with hypergraph containers. A monotone stepping-up argument then lifts the 3-uniform bound to 4-uniformity, and known recursions push it to higher uniformities.","feed_headline":"4-uniform Erdős–Rogers bound drops to (log n)^{o(1)}","feed_subtitle":"A new 3-uniform estimate plus a stepping-up argument settles a long-open hypergraph Ramsey problem.","key_machinery":"The proof combines three mechanisms. First, a robust auxiliary palette: a K_s-free graph F on q vertices such that every collection of at most s−1 sufficiently large vertex subsets contains a transversal clique; this palette is used to color pairs of a base set randomly, producing a 3-graph that is K_{s+1}^{(3)}-free by construction. Second, an auxiliary hypergraph Γ whose vertices are pairs (pair, color) and whose edges encode admissible color patterns that would force a K_s^{(3)}; a local coloring yields a K_s^{(3)}-free 3-graph exactly when its vertex set is independent in Γ. Third, a recursive application of a standard hypergraph container lemma counts the terminal containers and shows t","core_discovery":"The central claim is that, for every fixed s≥4, the 4-uniform Erdős–Rogers function f^{(4)}_{s,s+1}(n) is bounded by exp(O(log log n / log log log n)), hence is (log n)^{o(1)}. This means one can construct n-vertex 4-uniform hypergraphs with no K_{s+1}^{(4)} in which every K_s^{(4)}-free set has size at most that quantity. The key input is a new 3-uniform bound: f^{(3)}_{s,s+1}(n) ≤ C_s log n / log log n for every fixed s≥3. A further consequence is that for every fixed k≥5, f^{(k)}_{k+1,k+2}(n) = (log_{(k-3)} n)^{o(1)}, bringing the problem within one logarithmic iteration of the iterated-logarithm scale conjectured in the literature.","pith_inferences":["Editorial inference: the same palette-and-container counting strategy may extend to non-adjacent clique sizes t ≥ s+2, where the scales and local configuration hypergraphs change; the paper does not claim this extension.","Editorial inference: the exponential loss in the monotone stepping-up lemma suggests that a direct 4-uniform construction would be needed to reach the (log log n)^{O(1)} scale, which is a natural next target.","Editorial inference: if the q^{o(m)} factor in the container count could be removed or sharpened, the 3-uniform bound might yield better explicit constants or a stronger sub-logarithmic estimate.","Editorial inference: the robust-clique palette construction is likely reusable as a general tool for other hypergraph Ramsey-type counting problems, though no such reuse is explored here."],"forward_implications":["For every fixed s≥4, the 4-uniform Erdős–Rogers function f^{(4)}_{s,s+1}(n) is (log n)^{o(1)}, resolving the open 4-uniform problem.","The new 3-uniform estimate f^{(3)}_{s,s+1}(n) ≤ C_s log n / log log n improves the previous logarithmic upper bound by a factor of log log n.","For every fixed k≥5, f^{(k)}_{k+1,k+2}(n) ≤ exp(O_k(log_{(k-2)} n / log_{(k-1)} n)) = (log_{(k-3)} n)^{o(1)}, putting the iterated-logarithm conjecture within one logarithmic iteration.","In the case s=3, the 3-uniform estimate yields the inverse Ramsey bound r(K_4^{(3)}, K_m^{(3)}) ≥ 2^{Ω(m log m)}."],"fun_headline_variants":["Hypergraph Ramsey: 4-uniform sub-polylog bound","Erdős–Rogers: new 4-uniform (log n)^{o(1)}","3-uniform estimate powers 4-uniform breakthrough","Sub-polylog drop for 4-uniform Erdős–Rogers","Resolving Conlon–Fox–Sudakov: f^{(4)} = (log n)^{o(1)}"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The argument needs a standard hypergraph container lemma to apply at every recursive step to the auxiliary hypergraph, which requires a specific numerical balance between the palette size and the codegrees; the choice a = 1/(10h) leaves a margin of t^{-4/5}, and if that balance failed, the terminal container family would grow too large and the exponential saving over bad local colorings would disappear.","fun_headline_variants_meta":{"raw":{"variants":["Hypergraph Ramsey: 4-uniform sub-polylog bound","Erdős–Rogers: new 4-uniform (log n)^{o(1)}","3-uniform estimate powers 4-uniform breakthrough","Sub-polylog drop for 4-uniform Erdős–Rogers","Resolving Conlon–Fox–Sudakov: f^{(4)} = (log n)^{o(1)}"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000187,"raw_usage":{"total_tokens":1210,"prompt_tokens":831,"completion_tokens":379,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":575,"completion_tokens_details":{"reasoning_tokens":275}},"tokens_in":575,"tokens_out":379,"duration_ms":4070,"temperature":1.0,"reasoning_tokens":275,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T07:25:26.230704+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Enumerate, for a small concrete case such as s=3 and t around a few hundred, the number of local pair-colorings that produce a K_3^{(3)}-free 3-graph. If that count is not at most q^{(1−η)m} with η>0, or if the auxiliary palette F cannot be built with the claimed robust transversal property at q = t^a, then Lemma 3.5 fails and the main theorem collapses.","supporting_citations":[],"review_version":2}