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Transversal packings in families of percolated hypergraphs

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Pith's one-line read This paper proves that hypergraph systems slightly above the transversal Dirac threshold still contain a transversal $F$-factor after each member is independently thinned at the optimal edge probability $p = C n^{-1/d_1(F)-1} (\log…

desk verdict Sharp transversal F-factor threshold, with a fillable but real gap at the Riordan coupling step. read the letter →

arxiv 2507.12740 v1 pith:UW6YD6TT submitted 2025-07-17 math.CO

Jie Han , Jie Hu , Shunan Wei , Donglei Yang This is my paper
classification math.CO MSC 05C6505C7005C8005D40
keywords hypergraphsystemstransversalF-factorrandomsparsificationspreadmethodvertex-spreaddistributionDiracthresholdstrictly1-balancedhypergraphsrainbowfactors
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 a random-sparsification version of transversal Dirac-type theorems for hypergraphs. For any strictly $1$-balanced $k$-uniform hypergraph $F$ with $t$ edges, if each member of a hypergraph system is just above the degree threshold that guarantees a transversal $F$-factor, then after every member is independently thinned by retaining each edge with probability $p = C n^{-1/d_1(F)-1} (\log n)^{1/t}$, a transversal $F$-factor still exists with high probability. The exponent of $p$ is best possible up to the constant: below it, even the complete system typically has a vertex lying in no copy of $F$. The result therefore pins down, for a family of spanning rainbow packing problems, the exact rate at which percolation destroys the guaranteed structure. A companion theorem covers arbitrary $F$ at the slightly weaker rate $n^{-1/m_1(F)-1} \log n$.

What carries the argument

The carrying object is the expansion $F^*$: a $(k+1)$-uniform hypergraph obtained from $F$ by attaching one new color vertex to each edge, so that different edges receive different colors. Building the expanded hypergraph from a system $\mathbf H$ by adding the color set as a new vertex class, a transversal $F$-factor in $\mathbf H$ is exactly an $F^*$-factor in the expansion. The proof then (1) proves, by random clustering plus a cooperative perfect-matching theorem, a $(C/n)$-vertex-spread distribution on $F^*$-factor embeddings whenever the system is above $\delta^T_{F,d}$; (2) converts vertex-spreadness to a spread measure on perfect matchings of the $F^*$-complex; (3) applies the spread-threshold theorem to get a perfect matching in the percolated $F^*$-complex; and (4) uses a coupling theorem for strictly $1$-balanced hypergraphs. The strict $1$-balancedness of $F^*$ is verified directly: $d_1(F^*)=t/(s+t-1)$, so the coupling step applies.

What would settle it

Take the complete system $H_i = K_{sn}^{(k)}$ and test the boundary: Theorem 1.6 says $p = C n^{-1/d_1(F)-1} (\log n)^{1/t}$ works while $p = o(n^{-1/d_1(F)-1} (\log n)^{1/t})$ fails, because a sprinkling reduction to a random hypergraph below the $F$-factor threshold leaves some vertex uncovered. A simulation for $F = K_3$ in graphs, with $p$ near $n^{-5/3} (\log n)^{1/3}$, would exhibit the two regimes; if no fixed constant $C$ made the sufficiency side succeed, the sharpness claim would be wrong.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is Theorem 1.6: for every $\alpha>0$ and every strictly $1$-balanced $k$-uniform hypergraph $F$ on $s$ vertices with $t$ edges, there is a constant $C=C(k,s,t,\alpha)$ so that any $k$-graph system $\mathbf H=\{H_1,\dots,H_{tn}\}$ on $sn$ vertices with $\delta_d(H_i) \ge (\delta^T_{F,d}+\alpha)\binom{sn-d}{k-d}$ for all $i$ admits, with high probability, a transversal $F$-factor after independent random sparsification at $p = C n^{-1/d_1(F)-1} (\log n)^{1/t}$. Here a transversal $F$-factor means $n$ vertex-disjoint copies of $F$ using exactly one edge from each $H_i$, and $\delta^T_{F,d}$ is the infimum degree threshold for such factors in unpercolated systems. The paper shows the rate is sharp up to a constant by a sprinkling reduction to the known random-hypergraph $F$-factor threshold, and proves as an auxiliary step a spread version of the cooperative minimum-degree condition for perfect matchings in $k$-partite $k$-graphs, which is then used to build a $(C/n)$-vertex-spread distribution on embeddings of $F^*$-factors.

Load-bearing premise

The proof's load-bearing premise is that the transversal Dirac threshold keeps working for the constant-sized random subsystems created by clustering, and that the coupling step transferring random $F^*$-complexes to non-complete percolated hosts remains valid.

Editorial extensions

If this is right

  • For strictly $1$-balanced $F$, the probability bound is tight up to a constant: it matches, after sprinkling, the known random-hypergraph threshold for $F$-factors.
  • When $F$ is a graph and the transversal Dirac threshold coincides with the ordinary one, Theorem 1.6 gives a graph-system analogue of the known robust $F$-factor theorem.
  • For arbitrary $F$, Theorem 1.7 supplies a transversal $F$-factor at $p = C n^{-1/m_1(F)-1} \log n$, where $m_1(F)$ is the maximum $1$-density of a subgraph of $F$.
  • Taking each $H_i$ to be an independent random subhypergraph of one high-degree hypergraph $H$ yields a transversal $F$-factor in the collection at the same optimal rate (Corollary 5.5).
  • The vertex-spread construction implies that such systems contain $e^{(s+t-1)n\log n - O(n)}$ distinct transversal $F$-factors, giving an automatic counting statement.

Reading between the lines

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

  • If a single shared random thinning is used instead of independent percolation per member, the probability threshold should drop from $n^{-1/d_1(F)-1}$ to $n^{-1/d_1(F)}$ times a log-power; the paper leaves this as Question A and notes a positive answer for complete graph cliques.
  • The expansion-to-matching route is modular: the same recipe should transfer to other spanning rainbow structures, such as rainbow Hamilton cycles or powers of cycles, whenever a Dirac-type threshold and a spread distribution for the structure exist.
  • The explicit vertex-spread measure is an independent quantitative byproduct: it gives not just existence but a controlled distribution over rainbow factors, which may enable resilience or enumeration results beyond the percolation statement proven here.
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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

0 major / 6 minor

Summary. The paper studies transversal F-factors in percolated k-graph systems. Given a k-graph system H = {H_1,...,H_tn} on sn vertices with minimum d-degree slightly above the transversal Dirac threshold δ^T_{F,d}, it shows that after independent p-percolation of each H_i, with p = Ω(n^{-1/d_1(F)-1} (log n)^{1/t}), the system w.h.p. contains a transversal F-factor, and this p is best possible up to a constant. The proof expands F to a (k+1)-graph F* by adding a unique color to each edge, constructs a vertex-spread distribution on F*-factor embeddings via a random-clustering argument (Theorem 2.10 and Lemmas 3.1-3.2), converts it to a spread measure on perfect matchings of the F*-complex, applies the Frankston-Kahn-Narayanan-Park theorem to find a perfect matching in an independent percolation of the complex, and finally transfers this to the sparsified system via Riordan's coupling theorem. The paper also proves a spread version of Pikhurko's perfect-matching theorem for k-partite k-graphs (Theorem 2.7).

Significance. If the result holds, it is a natural and substantial extension of the recent robust F-factor theorems of Kelly, Müyesser and Pokrovskiy, and of Joos, Lang and Sanhueza-Matamala, to transversal settings, with an edge probability that is best possible up to constants and matches the random-graph threshold for the union of the colors. The proof methodology is current and technically involved: the random-clustering lemmas contain full concentration arguments, Claim 2.13 correctly proves strict 1-balancedness of the expanded hypergraph F*, and the construction of vertex-spread distributions for transversal factors is new. The paper is honest about using the external Dirac threshold δ^T_{F,d} as a black box, and the central claim is not assumed in the proof. The main theorem yields explicit, falsifiable threshold predictions and extends the reach of the spread method to transversal packing problems.

minor comments (6)
  1. [§2.3, proof of Theorem 1.6] The sentence applying Theorem 2.12 to transfer a perfect matching in H_{F*}(π) to an F*-factor in H(p) is too terse, because Theorem 2.12 is stated only for the complete binomial random hypergraph. Please add a short justification: apply Theorem 2.12 to the complete (k+1)-graph on V(H) ∪ C(H) with F = F*, then restrict the independent percolation on the full F*-complex to the subcomplex H_{F*}; every surviving H-copy is then realized in H(p), since the coupled binomial (k+1)-graph induces independent random k-graphs on the color classes and H(p) is the intersection of H with those. This restricted-host coupling is valid and standard, but it should be stated explicitly.
  2. [§1.4, notation] The symbol GF(n,p) is first defined as the F-complex of G(k)(n,p), but in Theorem 2.12 it is used to mean an independent percolation on the F-complex of the complete k-graph, where each copy of F is retained independently with probability p. Please disambiguate the definition; otherwise the statement of Theorem 2.12 is tautological because the F-complex of G is determined by G.
  3. [§3.2, proof of Lemma 3.1, property (3)] The displayed bound on P[∩ D^1_{i,j}] uses cluster sizes s(C-1) and t(C-1), which is correct only for i ≥ 2; for the exceptional cluster U_1 (resp. W_1) the size is about sC(C-1) (resp. tC(C-1)), giving a larger probability by a factor depending on C. The conclusion of property (3) is still true if C' is chosen suitably larger than C, but the proof should say this explicitly.
  4. [§3.1, deduction of Theorem 2.8] The proof of Theorem 2.8 from Lemma 3.2 is omitted with the justification that it follows the same argument as the proof of Theorem 2.10 from Lemma 3.1. Since Theorem 2.8 is used in the proof of Theorem 2.7, which is itself used in Lemma 3.1, please include a detailed proof or at least a precise outline of this deduction.
  5. [§2.3, proof of Theorem 1.6 and Lemma 3.1] When applying Definition 1.5 to the constant-size sub-systems produced by the random clustering, the paper should explicitly state that the constant C is chosen larger than the hidden n0 in Definition 1.5; this is implicit in the hierarchy 1/C ≪ α, 1/k, 1/s, 1/t, but it is a uniformity assumption that deserves to be stated.
  6. [§3.2, notation in the proof of Lemma 3.1] The events D_{i,j}, D^1_{i,j}, and D^2_{i,j} are indexed by both a vertex index i and a color index j, which makes the later unions over i ∈ [r], j ∈ [ℓ] awkward; a cleaner indexing (separate vertex events and color events) would help readability. Also, in Corollary 5.5 the notation H(p) is reused for a single hypergraph and for the system; please distinguish the two usages.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the proof reduces to an external Dirac threshold and independent spread/coupling machinery; the only flagged concern is a non-circular gap in applying Riordan's coupling to a restricted host.

full rationale

The derivation is self-contained in the sense relevant to circularity. Theorem 1.6 is conditional on the external p=1 transversal Dirac threshold delta^T_{F,d} (Definition 1.5), which is used as a black-box hypothesis rather than derived from the random conclusion; applying it to constant-size clusters is a legitimate large-C choice, not a circular reduction. The spread machinery (Theorems 2.7, 2.8, 2.10, Proposition 2.4, Lemmas 3.1 and 3.2) constructs vertex-spread measures from the Dirac input and Pikhurko-type matching arguments, without assuming the target percolation statement. The FKN theorem (2.11) and Riordan's coupling (2.12) are external results; the only concern is that Theorem 2.12 is quoted for the complete binomial random hypergraph while it is applied to the F*-complex of an arbitrary Dirac system, so the one-sentence transition in the proof of Theorem 1.6 is a possible proof gap, but not circularity, because 'H(p) contains an F*-factor' is not among the hypotheses of the argument. Self-citation [19] appears only in the concluding remarks and is not load-bearing. Hence no step reduces to its own input; score 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 1 invented entities

No numbers are fitted to data. The constants C, C', C_{2.2}, etc. are universal existence constants chosen large in the hierarchy 1/n ≪ 1/C' ≪ 1/C ≪ α; they are not free parameters in the sense of fitted or ad hoc values that the theorem's conclusion depends on.

assumptions (5)
  • standard math FKNP spread theorem (Theorem 2.11)
    Used to turn the (C n^{-(s+t-1)})-spread measure on perfect matchings of H_{F*} into a w.h.p. perfect matching in H_{F*}(π). Published theorem [15].
  • standard math Riordan coupling theorem (Theorem 2.12)
    Converts a perfect matching in the random F*-complex into an F*-factor in the randomly sparsified host H(p); requires F* strictly 1-balanced, proved in Claim 2.13.
  • standard math Pham-Sah-Sawhney-Simkin spread distribution on dense bipartite perfect matchings (Theorem 2.2)
    Used in both redistribution rounds of Lemma 3.2 to sample spread perfect matchings in auxiliary bipartite graphs of minimum degree 3/4(m-1). Source is [38], an arXiv preprint.
  • standard math Kelly-Müyesser-Pokrovskiy vertex-spread to spread conversion (Proposition 2.4)
    Converts the (C/n)-vertex-spread distribution on F*-factor embeddings into a (C'/n^{s+t-1})-spread measure on perfect matchings of H_{F*}. Published in [26].
  • domain assumption Definition and existence of the transversal Dirac threshold δ^T_{F,d} with uniform n0
    Theorem 1.6 and Theorem 2.10 are conditional on the p=1 transversal Dirac threshold; the proof applies it to subsystems of constant size sC, which requires C to exceed the 'sufficiently large n' in the definition.
invented entities (1)
  • The expanded hypergraph F* and the color-augmented system H
    purpose: Reduces the transversal F-factor problem to finding an F*-factor in a single (k+1)-graph, enabling the spread method.
    This is a proof construction, not an empirical entity. It is fully defined in Section 2.2 and its properties (strict 1-balancedness) are proved in Claim 2.13.

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Pith. "Pith review of Transversal packings in families of percolated hypergraphs." pith.science (2026). https://pith.science/paper/UW6YD6TT

@misc{pith2026250712740,
  author       = {Pith},
  title        = {Pith review of: Transversal packings in families of percolated hypergraphs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UW6YD6TT}},
  note         = {Machine review of arXiv:2507.12740}
}
abstract

Let $F$ be a strictly $1$-balanced $k$-graph on $s$ vertices with $t$ edges and $\delta_{F,d}^T$ be the infimum of $\delta>0$ such that for every $\alpha>0$ and sufficiently large $n\in \mathbb{N}$, every $k$-graph system $\mathbf H=\{H_{1}, H_{2}, \dots ,H_{tn}\}$ on the same $sn$ vertices with $\delta_d(H_i)\ge (\delta+\alpha)\binom{sn-d}{k-d}$, $i\in [tn]$ contains a transversal $F$-factor, that is, an $F$-factor consisting of exactly one edge from each $H_i$. In this paper we prove the following result. Let $\mathbf{H} =\{H_{1}, H_{2}, \dots ,H_{tn}\}$ be a $k$-graph system where each $H_{i}$ is an $sn$-vertex $k$-graph with $\delta_d(H_i)\ge (\delta_{F,d}^T+\alpha)\binom{sn-d}{k-d}$. Then with high probability $\mathbf{H}(p) :=\{H_{1}(p), H_{2}(p), \dots ,H_{tn}(p)\}$ contains a transversal $F$-factor, where $H_i(p)$ is a random subhypergraph of $H_i$ and $p=\Omega(n^{-1/d_1(F)-1}(\log n)^{1/t})$. This extends a recent result by Kelly, M\"{u}yesser and Pokrovskiy, and independently by Joos, Lang and Sanhueza-Matamala. Moreover, the assumption on $p$ is best possible up to a constant. Along the way, we also obtain a spread version of a result of Pikhurko on perfect matchings in $k$-partite $k$-graphs.

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