{"id":"3182a707-202e-4adf-a66b-311d4e17bad7","arxiv_id":"2505.05606","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Every large 3-uniform hypergraph whose vertex count is a multiple of five and whose minimum codegree is at least 2n/5 admits a perfect tiling by the generalized triangle, and this threshold is best possible.","lead":"This paper finds the exact minimum codegree condition for tiling any sufficiently large 3-uniform hypergraph with copies of the generalized triangle, a five-vertex three-edge shape: every pair of vertices must lie in at least 2n/5 triples. It also proves an asymptotically optimal condition for the rainbow version of the same tiling problem.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No load-bearing concern for Theorem 1.1; the reader's worry about the T3 cases in Lemma 4.2 does not land, but the 5-divisibility reduction in Lemma 4.2 contains a fixable averaging slip.","rationale":"The reader's weakest assumption pointed to the construction of T1, T2, T3 in Lemma 4.2, especially the second T3 subcase. I examined those constructions and found them valid: the intersection bound for W' follows from a union bound, and the successive choices have enough slack, so the domination and Farkas contradiction go through. The actual weak point I found is elsewhere in the same lemma, namely the blow-up reduction that is meant to extend Lemma 4.2 from 5|n to all n. As written, the projected fractional tiling is not perfect, so Lemma 4.2 and Lemma 1.5 are not proved for arbitrary n. However, Theorem 1.1 only needs the 5|n case: the absorbing set has size divisible by 5, so H' has n' divisible by 5, and the 5|n proof of Lemma 4.2 is independent of the flawed reduction. The flaw is therefore non-central and easily fixable by correcting the projection factor or by restricting Lemmas 4.2 and 1.5 to n divisible by 5. For this reason I do not change the reader's ACCEPT verdict, but I would ask the authors to repair the reduction before publication.","tokens_in":24956,"tokens_out":42686,"duration_ms":433371,"concrete_test":"Recompute the projection in the first paragraph of Lemma 4.2 on a base case: take a perfect T-tiling of H, lift each copy to the 5-blow-up with uniform weight 1/5^4 on each of the 5^5 liftings, and evaluate the load on a fixed original vertex u under the paper's averaging rule; it equals 1/5^4, not 1. Then rerun the proof of Lemma 1.5 with n restricted to multiples of 5 to check that the flawed general-n reduction is never needed along the path used by Theorem 1.1.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I checked the divisible-n core of Lemma 4.2, which is the part used by Theorem 1.1. The constructions of T1, T2 and T3 are internally consistent: the bound '3n/5+αn+4εn≤βn' must be read as '3n/5+αn+4εn≤3n/5+βn', and with that correction the domination arguments for V1, V2 and V3 go through, including the two T3 subcases. The real defect I find is in the opening reduction of Lemma 4.2. It claims that projecting a perfect B'-avoiding fractional T-tiling w' from the 5-blow-up H' by averaging over all 5^5 liftings of each copy T* gives a perfect fractional T-tiling in H. For a fixed original vertex u, the projected load is (1/5^5) * sum_{i=1}^5 w'(u_i) = 1/5^4, not 1. The correct projection is to sum the weights of all liftings and divide by 5, not by 5^5. Thus Lemma 4.2 as stated for all n is not proved by the given reduction. This does not threaten Theorem 1.1: in the proof of the main theorem, Lemma 1.5 is applied only to H' with |V(H')| divisible by 5, because the absorbing set A has size divisible by 5, and the 5|n case of Lemma 4.2 is proved directly after the reduction and is sound. So the central claim stands, with a small revision needed in the statements or proofs of Lemmas 4.2 and 1.5.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves an exact minimum codegree threshold for perfect tilings by the generalized triangle T: for all sufficiently large n divisible by 5, every 3-uniform hypergraph H on n vertices with δ(H) ≥ 2n/5 contains a perfect T-tiling, and the constant 2/5 is best possible, as shown by H_ext with |A|=2n/5−1 and all edges meeting A. The proof combines an extremal case (Section 2), an absorbing lemma (Section 3), and an almost-perfect tiling lemma (Section 4) obtained from a fractional T-tiling via Farkas' lemma and a Pippenger–Spencer matching result. Section 5 derives an asymptotically optimal rainbow version from a theorem of Lang.","tokens_in":25277,"tokens_out":21768,"duration_ms":207613,"significance":"The result is a genuine step forward: K_4^3 and K_4^3−e were previously the only non-tripartite 3-graphs for which an optimal codegree threshold for perfect tilings was known, and T is the first such graph on more than four vertices. The lower-bound example is simple and correct, and the proof is detailed, with the absorbing and extremal components carefully structured. The Farkas-lemma case analysis in Section 4 is a novel technique with potential for further applications. If the Section 4 statements are corrected as described below, the main theorem is sound; the rainbow corollary is conditional on the unpublished preprint [33].","major_comments":[{"comment":"The reduction to the case 5|n is invalid. In the 5-blow-up H', a pair u_i v_j with u,v∈V and uv∉E(H) is contained only in the triples u_i u_{j'} v_j with j'≠i, so its codegree is 4, whereas 5δ(H) is about 2n; hence δ(H') ≥ 5δ(H) is false and the case 5|n cannot be applied to H'. Even if the codegree were preserved, the projection is mis-normalized: averaging w' over all 5^5 liftings of a copy T* assigns load 1/5^4 to each vertex of H, not 1; the correct projection is to sum the weights of all liftings and divide by 5. The remainder of the proof after 'so assume that 5 divides n' is self-contained, so the cleanest repair is to state Lemmas 4.2, 4.3, and 1.5 only for 5|n, which is the only case used in the proof of Theorem 1.1.","section":"Section 4, Lemma 4.2, first paragraph"},{"comment":"The application of Lemma 4.2 in the proof of Lemma 4.3 is not justified as written: from W ≥ 1/(εn) one gets ∆(B) ≤ 4/W ≤ 4εn, while Lemma 4.2 requires ∆(B) ≤ εn. The argument can be repaired by applying Lemma 4.2 with parameter 4ε (or a rescaled constant), so this is a local but necessary correction.","section":"Section 4, Lemma 4.3"}],"minor_comments":[{"comment":"In the construction of T1 the displayed inequality '3n/5+αn+4εn≤βn' is false; it should read '≤3n/5+βn' for the subsequent domination by V1 to hold.","section":"Section 4, proof of Lemma 4.2, T1 construction"},{"comment":"The proof of δ_c=0 says 'Let H1,H2 be graphs'; these should be 3-graphs. It would also be clearer to state explicitly that for n large the condition δ(H_i)≥(δ+µ)n with δ=0 yields δ(H_i)≥3, so the construction applies.","section":"Section 5, proof of Theorem 1.6"},{"comment":"The notation '5||S|' is nonstandard; consider writing '5 divides |S|' for clarity.","section":"Section 1.3, proof of Theorem 1.1"}],"recommendation":"major_revision","confidential_remarks":"For the editor: the main theorem is not endangered by the Section 4 problems because Lemma 1.5 is only applied with n' divisible by 5; however, the statements of Lemmas 4.2, 4.3, and 1.5 need to be restricted or the reduction fixed. The rainbow part depends on the unpublished preprint [33], so its correctness is contingent on the status of that work."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Worth knowing about this paper: it settles the exact minimum codegree threshold for perfect T-tilings in 3-graphs, where T is the 5-vertex generalized triangle, and the threshold is 2n/5. This is the first exact threshold for a non-tripartite 3-graph with more than four vertices, so it is a real step forward, not just incremental.\n\nCredit where it's earned. The extremal example is simple and correct. The proof is solid machinery: absorption (Lo-Markström, Han-Treglown) plus a Farkas lemma argument that uses the structure of H to choose dominated copies of T. That structural twist in Lemma 4.3/4.2 is genuinely new as far as I know. The extremal case is handled carefully via an auxiliary 5-partite 5-graph and a Daykin-Häggkvist matching theorem.\n\nSoft spots, in proportion. The stress-test note I saw found a real slip in the opening reduction of Lemma 4.2. The paper claims that projecting a fractional tiling from the 5-blow-up by averaging over all 5^5 liftings gives a fractional tiling in H; the arithmetic gives vertex-load 1/5^4, not 1. To make the projection work, you divide by 5, not 5^5. So Lemma 4.2 as stated for all n is not established by the written proof. That is not a fatal flaw for Theorem 1.1: the main theorem only applies Lemma 1.5 to H' with 5 dividing n, because the absorbing set has size divisible by 5, and the 5|n case of Lemma 4.2 is proved directly and looks sound. Still, the statements of Lemmas 4.2 and 1.5 need adjustment or a corrected reduction. Also, the rainbow corollary depends on an unpublished preprint of Lang; that is a dependency to flag rather than a flaw.\n\nThe reader's worry about the T3 construction in Lemma 4.2 does not land. I checked the two subcases; the domination argument goes through once you read the '3n/5 + βn' bound correctly. So the central argument holds.\n\nWho this is for: people working in hypergraph tilings and Dirac-type thresholds. It is a significant result and deserves serious refereeing. My verdict: engage with it, send it out. The averaging slip needs fixing, but the main theorem stands.","headline":"First exact codegree threshold for perfect tilings with a non-tripartite 3-graph on more than four vertices; the main theorem is sound, though Lemma 4.2 has a fixable averaging slip.","tokens_in":25835,"tokens_out":1824,"would_cite":true,"duration_ms":17708,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C65","05C70"],"pacs":[],"model":"deepseek-v4-flash","headline":"For large n divisible by 5, every 3-uniform hypergraph with minimum codegree at least 2n/5 has a perfect tiling by the generalised triangle, and the bound is optimal.","keywords":["perfect tilings","minimum codegree","3-uniform hypergraphs","generalised triangle","rainbow tilings","absorption method","fractional tilings","Farkas' lemma"],"falsifier":"An infinite family of 3-graphs $H_n$ with $5\\mid n$, $\\delta(H_n)\\ge 2n/5$, and no perfect $T$-tiling would refute the main theorem; since every $\\gamma$-extremal such graph is shown tileable, such a family would have to be non-extremal. A concrete way to search is to perturb the extremal example so that every copy of $T$ still needs two vertices from a blocker set of size $2n/5-1$ while every $3n/5$-vertex set induces density above $\\gamma$.","tokens_in":24750,"feed_emoji":"🔺","tokens_out":18834,"duration_ms":167335,"temperature":0.7,"pith_summary":"The paper establishes an exact minimum-codegree threshold for a hypergraph tiling problem. For every sufficiently large $n$ divisible by 5, any 3-uniform hypergraph $H$ on $n$ vertices with minimum codegree $\\delta(H) \\ge 2n/5$ admits a perfect tiling by copies of the generalised triangle $T$: the five-vertex 3-graph with edges $abc$, $abd$, and $cde$. The constant is best possible, because the hypergraph whose edges are all triples meeting a set of $2n/5-1$ vertices has minimum codegree $2n/5-1$ and no perfect $T$-tiling. This makes $T$ the first non-tripartite 3-uniform hypergraph with more than four vertices for which the optimal perfect-tiling threshold is known exactly rather than asymptotically. The paper also proves an asymptotically optimal rainbow version: if each of $3n/5$ hypergraphs on a common vertex set has codegree at least $(2/5+\\varepsilon)n$, then a perfect rainbow $T$-tiling exists.","feed_headline":"Codegree 2n/5 guarantees perfect generalised-triangle tilings","feed_subtitle":"The bound is exact: one fewer codegree blocks all tilings, and rainbow versions match it asymptotically.","key_machinery":"The load-bearing objects are the generalised triangle $T$ (the 3-graph with five vertices and edges $abc$, $abd$, $cde$), a fractional $T$-tiling (a weighting of copies of $T$ so every vertex has total weight 1), and the vector $a\\in\\mathbb{R}^n$ produced by Farkas' lemma when no perfect fractional $T$-tiling exists. Ordering vertices by $a_1\\le\\cdots\\le a_n$ defines three families of five-vertex sets, and the central technical step is to find three copies $T_1,T_2,T_3$ of $T$, each $B$-avoiding for a small graph $B$ of forbidden pairs, each dominated by the corresponding family. Domination means the $i$-th smallest vertex of the copy is no larger than the $i$-th smallest vertex of the set, so $a\\cdot\\mathbf{1}_{T_i}\\le a\\cdot\\mathbf{1}_V$ for every $V$ in that family. Summing the inequalities over the three families contradicts $a\\cdot\\mathbf{1}<0$, forcing a perfect fractional $T$-tiling whose pair weights are bounded by $1/(\\varepsilon n)$; a matching theorem for multi-hypergraphs then turns this into an almost-perfect tiling, and absorption upgrades it to a perfect one.","core_discovery":"The central claim is that $\\delta(H)\\ge 2n/5$ is the exact barrier for perfect $T$-tilings in 3-graphs. The proof is split into an extremal case, where $H$ contains a set of $3n/5$ vertices inducing low density and a tiling is completed by finding a perfect matching in a dense auxiliary 5-partite 5-graph, and a non-extremal case, handled by absorption. In the non-extremal case the paper constructs a perfect fractional $T$-tiling with bounded pair weights via Farkas' lemma, then converts it to an almost-perfect tiling and uses a small absorber to finish. The matching threshold cannot be lowered: the hypergraph $H_{\\text{ext}}$ with parts of sizes $2n/5-1$ and $3n/5+1$, containing all triples that meet the smaller part, has minimum codegree $2n/5-1$ and no perfect $T$-tiling.","pith_inferences":["The authors conjecture that the 'sufficiently large' condition is unnecessary; if that is right, the exact $2n/5$ formula holds for every $n$ divisible by 5, and the extremal example is the only obstruction at every scale.","Because the colour-covering threshold for $T$ is zero, the rainbow threshold equals the ordinary tiling threshold; comparing this with hypergraphs whose colour-covering threshold is positive would quantify when rainbow tilings become strictly harder.","The structure-dependent Farkas step suggests a general template for non-tripartite $F$: identify the extremal construction, use the separation vector to define order-based vertex classes, and construct one dominating $F$-copy per class with cases chosen from neighbourhood intersections.","Exact-cover searches on small multiples of 5 could test the no-large-$n$ conjecture and would expose the first possible non-extremal obstruction if the threshold formula fails before $n_0$."],"forward_implications":["For all sufficiently large $n$ divisible by 5, any 3-uniform hypergraph with minimum codegree at least $2n/5$ has a perfect tiling by the generalised triangle.","The threshold $2n/5$ is sharp: the constructed extremal hypergraph has codegree $2n/5-1$ and no perfect $T$-tiling.","The rainbow version holds with the same constant asymptotically: for every $\\varepsilon>0$, a family of $3n/5$ hypergraphs with common vertex set and each codegree at least $(2/5+\\varepsilon)n$ admits a perfect rainbow $T$-tiling.","This is the first exact perfect-tiling threshold for a non-tripartite 3-uniform hypergraph with more than four vertices, placing the generalised triangle alongside $K_4^3$ and $K_4^3-e$ as cases where the codegree threshold is fully determined."],"supporting_citations":[{"why":"Supplies the perfect-matching theorem for dense k-partite k-graphs used to complete the extremal-case tiling.","marker":"[10]"},{"why":"Supplies the absorption lemma that upgrades an almost-perfect tiling to a perfect one.","marker":"[36]"},{"why":"Supplies the partition lemma that turns pairwise linkedness into a partition of the vertex set into closed parts.","marker":"[21]"},{"why":"Supplies the lattice lemma that propagates closure from the parts to the whole vertex set.","marker":"[22]"},{"why":"Supplies the nearly-regular hypergraph matching theorem that converts a fractional tiling with small pair weights into an almost-perfect integral tiling.","marker":"[41]"},{"why":"Supplies the identity relating the tiling, rainbow, and colour-covering thresholds used for the rainbow corollary.","marker":"[33]"}],"fun_headline_variants":["Codegree 2n/5 exactly forces perfect tilings","Sharp codegree 2n/5 for generalized triangle tilings","Perfect tilings need codegree 2n/5, no less","Exact codegree barrier for triangle tilings found","Codegree 2n/5 pins down generalized triangle tilings"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The non-extremal proof depends on a case split asserting that, for every ordering of the vertices by the separation vector, three prescribed five-vertex families can each be dominated by an actual copy of the generalised triangle that avoids a small set of forbidden pairs; if any one of the three constructions cannot be realized, the contradiction forcing a perfect fractional tiling collapses.","fun_headline_variants_meta":{"raw":{"variants":["Codegree 2n/5 exactly forces perfect tilings","Sharp codegree 2n/5 for generalized triangle tilings","Perfect tilings need codegree 2n/5, no less","Exact codegree barrier for triangle tilings found","Codegree 2n/5 pins down generalized triangle tilings"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000446,"raw_usage":{"total_tokens":2194,"prompt_tokens":823,"completion_tokens":1371,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":439,"completion_tokens_details":{"reasoning_tokens":1283}},"tokens_in":439,"tokens_out":1371,"duration_ms":9895,"temperature":1.0,"reasoning_tokens":1283,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T23:03:52.461762+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"An infinite family of 3-graphs $H_n$ with $5\\mid n$, $\\delta(H_n)\\ge 2n/5$, and no perfect $T$-tiling would refute the main theorem; since every $\\gamma$-extremal such graph is shown tileable, such a family would have to be non-extremal. A concrete way to search is to perturb the extremal example so that every copy of $T$ still needs two vertices from a blocker set of size $2n/5-1$ while every $3n/5$-vertex set induces density above $\\gamma$.","supporting_citations":[{"cited_title":"Daykin and R","cited_arxiv_id":null,"evidence_quote":"Supplies the perfect-matching theorem for dense k-partite k-graphs used to complete the extremal-case tiling."},{"cited_title":"Lo and K","cited_arxiv_id":null,"evidence_quote":"Supplies the absorption lemma that upgrades an almost-perfect tiling to a perfect one."},{"cited_title":"Han and A","cited_arxiv_id":null,"evidence_quote":"Supplies the partition lemma that turns pairwise linkedness into a partition of the vertex set into closed parts."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the lattice lemma that propagates closure from the parts to the whole vertex set."},{"cited_title":"Pippenger and J","cited_arxiv_id":null,"evidence_quote":"Supplies the nearly-regular hypergraph matching theorem that converts a fractional tiling with small pair weights into an almost-perfect integral tiling."}],"review_version":1}