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Perfect tilings of 3-graphs with the generalised triangle

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2505.05606 v1 pith:UPOCRUEU submitted 2025-05-08 math.CO

Candida Bowtell , Amarja Kathapurkar , Natasha Morrison , Richard Mycroft This is my paper · ORCID
classification math.CO MSC 05C6505C70
keywords perfecttilingsminimumcodegree3-uniformhypergraphsgeneralisedtrianglerainbowabsorptionmethodfractionalFarkas'lemma
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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 $\gamma$.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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 $n_0$.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 3 minor

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.

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 (2)
  1. [Section 4, Lemma 4.2, first paragraph] 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.
  2. [Section 4, Lemma 4.3] 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.
minor comments (3)
  1. [Section 4, proof of Lemma 4.2, T1 construction] 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.
  2. [Section 5, proof of Theorem 1.6] 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.
  3. [Section 1.3, proof of Theorem 1.1] The notation '5||S|' is nonstandard; consider writing '5 divides |S|' for clarity.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; Theorem 1.1 rests on external prior results and the rainbow corollary applies Lang's theorem independently.

full rationale

The main derivation is self-contained against external benchmarks. Lemma 1.3 uses the Daykin–Häggkvist matching theorem; Lemma 1.4 is obtained from Lo–Markström's absorbing theorem together with Han–Treglown and Han–Zang–Zhao; Lemma 1.5 is derived from Farkas' lemma and Pippenger–Spencer. None of these inputs states Theorem 1.1 or is fitted to its conclusion, and the extremal lower bound H_ext is constructed independently and verified directly. The rainbow corollary computes δ_c=0 by an explicit construction and δ_t=2/5 from Theorem 1.1 plus H_ext, then applies Lang's external relation δ_r=max(δ_c,δ_t); this is a legitimate deduction, not a renaming or a self-citation. The paper's citations to the authors' own previous work (e.g., Mycroft [39], Keevash–Mycroft [25]) are contextual and not load-bearing for Theorem 1.1. The only salient defect, an averaging slip in the blow-up reduction at the opening of Lemma 4.2, is a non-circular technical gap in the case where 5 does not divide n; the divisible case actually used in the proof of Theorem 1.1 is proved directly, so the circularity verdict is unaffected.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The central claim rests on a sequence of published theorems (Daykin-Haggkvist, Lo-Markström, Han-Treglown, Han-Zang-Zhao, Pippenger-Spencer) plus Farkas' lemma; these are external, not introduced ad hoc. The rainbow corollary depends on Lang's Theorem 5.1 from an arXiv preprint. No free parameters or invented entities appear.

assumptions (7)
  • standard math Farkas' lemma (Lemma 4.1)
    Used in Lemma 4.2 to derive the existence of a separating vector a from the absence of a perfect B-avoiding fractional T-tiling; standard convex geometry.
  • domain assumption Daykin-Haggkvist theorem (Theorem 2.1)
    Sufficient condition for a perfect matching in a k-partite k-graph; used in the extremal case to find a perfect matching in the auxiliary 5-graph J.
  • domain assumption Lo-Markström absorption lemma (Theorem 3.1)
    Provides an absorbing set from (η,t)-closure; used to prove Lemma 1.4.
  • domain assumption Han-Treglown partition theorem (Theorem 3.3)
    Partitions V(H) into closed parts from linkedness conditions; used in Lemma 3.2.
  • domain assumption Han-Zang-Zhao lattice theorem (Theorem 3.5)
    Lifts closure from parts to the whole vertex set using index vectors and the lattice L; used in Lemma 3.2.
  • domain assumption Pippenger-Spencer matching theorem (Theorem 4.4) and Corollary 4.5
    Converts near-perfect fractional matchings into large integer matchings; used in Lemma 1.5.
  • domain assumption Lang's theorem δ_r = max(δ_c, δ_t) (Theorem 5.1)
    Relates the rainbow tiling threshold to the tiling and colour covering thresholds; used for Theorem 1.6. This is a dependency on an arXiv preprint.

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Pith. "Pith review of Perfect tilings of 3-graphs with the generalised triangle." pith.science (2026). https://pith.science/paper/UPOCRUEU

@misc{pith2026250505606,
  author       = {Pith},
  title        = {Pith review of: Perfect tilings of 3-graphs with the generalised triangle},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UPOCRUEU}},
  note         = {Machine review of arXiv:2505.05606}
}
abstract

We establish a best-possible minimum codegree condition for the existence of a perfect tiling of a $3$-uniform hypergraph $H$ with copies of the generalised triangle $T$, which is the 3-uniform hypergraph with five vertices $a, b, c, d, e$ and three edges $abc$, $abd$, $cde$. We also give an asymptotically-optimal minimum codegree condition for the rainbow version of the problem.

Figures

Figures reproduced from arXiv: 2505.05606 by the authors.

Figure 1
Figure 1. The generalised triangle 3-graph T. 1.1 Perfect tilings in graphs and hypergraphs The general question of determining whether a graph or hypergraph H contains some given spanning structure – such as a perfect matching, a Hamilton cycle or a perfect F-tiling – has been one of the most-studied topics in graph theory since the origins of the subject. For example, two classic results in this area are Tutte’s theorem [47… view at source ↗

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Positive codegree thresholds for perfect matchings in hypergraphs

    math.CO 2025-05 accept novelty 7.0 of 10

    For every k ≥ 3, a large k-uniform hypergraph with minimum positive codegree at least (k−1)/k n − (k−2) and no isolated vertices must contain a perfect matching, and this bound is best possible.

Reference graph

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