By Peter Keevash

The authors advance a conception for the life of excellent matchings in hypergraphs below really common stipulations. Informally conversing, the obstructions to ideal matchings are geometric, and are of 2 specific varieties: 'space limitations' from convex geometry, and 'divisibility obstacles' from mathematics lattice-based buildings. To formulate specific effects, they introduce the atmosphere of simplicial complexes with minimal measure sequences, that is a generalisation of the standard minimal measure . They confirm the basically very best minimal measure series for locating a nearly excellent matching. moreover, their major end result establishes the soundness estate: less than an identical measure assumption, if there's no ideal matching then there needs to be an area or divisibility barrier. this permits using the steadiness procedure in proving special effects. in addition to recuperating past effects, the authors observe our conception to the answer of 2 open difficulties on hypergraph packings: the minimal measure threshold for packing tetrahedra in 3-graphs, and Fischer's conjecture on a multipartite type of the Hajnal-Szemeredi Theorem. the following they end up the precise consequence for tetrahedra and the asymptotic end result for Fischer's conjecture; because the detailed consequence for the latter is technical they defer it to a next paper

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It also implies |Uf − Uf | ≤ DF kdα for any f, f ∈ F . Now, by deﬁnition of GF and invariance of F under Symk , for any i, i ∈ [r] and j, j ∈ [k] such that ii ∈ GF , we can choose f ∈ F with f (j) = i and f (j ) = i , and let f ∈ F be obtained from f by transposing the values on j and j . Then |Uf − Uf | = |(ui,j −ui,j )−(ui ,j −ui ,j )| ≤ DF kdα. 2. 1, we say that an edge e ∈ M is good if a · χ(e ) ≤ U + d α. 2 (i) edge of M . Note that if vi,j √ gives 0 ≤ (a − a) · χ(e ) ≤ d α, so ai,j − ai,j < d α.

Since i ∈ [r] and u, v ∈ Vi were arbitrary, it follows that (Jk , M ) is (B, C)-irreducible with respect to P . For F (J) ≥ n/k − αn for the ﬁnal statement we claim that δ + (D1 (Jk , M )[Vq ]) ≥ δk−1 each q ∈ [r]. Indeed, consider any v ∈ Vq , and let e be the edge of M containing v. Then we can write e = {v1 , . . , vk } where vk = v and vi ∈ Vf (i) for i ∈ [k] for some f f ∈ F . By assumption e \ {v} ∈ J, so there are at least δk−1 (J) vertices u ∈ Vq such that {u} ∪ e \ {v} ∈ J. For each such u, ({{u} ∪ e \ {v}}, {e }) is a simple (u, v)-transferral in (Jk , M ) of size one, so this proves the claim.

Uk . Then there exists a P-partite k-complex G on U such that Gk ⊆ G, G is ε-regular, d[k] (G ) ≥ d/2, d(G ) ≥ da , and Z = Z ∩ Gk has |Z | ≤ ν 1/3 |Gk |. Proof. To ﬁnd G we select a suitable cell of P ∗ , which we recall is the ak bounded Q-partition k-complex formed from P by weak equivalence. Note that one of the partitions forming P ∗ is a partition of K[k] (U ) into cells C1 , . . , Cs , where s ≤ ak . So at most dnk1 /3 edges of K[k] (U ) lie within cells Ci such that |Ci | ≤ dnk1 /(3ak ).