By Peter Keevash

The authors enhance a idea for the lifestyles of ideal matchings in hypergraphs below really normal stipulations. Informally talking, the obstructions to ideal matchings are geometric, and are of 2 unique varieties: 'space obstacles' from convex geometry, and 'divisibility obstacles' from mathematics lattice-based structures. To formulate special effects, they introduce the surroundings of simplicial complexes with minimal measure sequences, that is a generalisation of the standard minimal measure . They be certain the basically very best minimal measure series for locating a nearly excellent matching. additionally, their major outcome establishes the steadiness estate: below an analogous measure assumption, if there's no excellent matching then there needs to be an area or divisibility barrier. this enables using the soundness strategy in proving detailed effects. along with getting better past effects, the authors follow our idea 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 kind of the Hajnal-Szemeredi Theorem. right here they end up the precise outcome for tetrahedra and the asymptotic outcome for Fischer's conjecture; because the particular end result for the latter is technical they defer it to a next paper

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**Extra resources for A geometric theory for hypergraph matching**

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Suppose that 1/n 1/ 1/B, 1/C α α β 1/DF , 1/r, 1/k. Let V be a set of rn vertices, and let P partition V into r parts V1 , . . , Vr each of size n. Suppose F is a (k, r)-uniform connected allocation with |F | ≤ DF , and (J, M ) is a P F -partite matched k-system on V such that M α represents F . Suppose also that k−2 1 (i) δ F (J) ≥ n, k−1 k − α n, k − α n, . . , k − α n , and (ii) for any p ∈ [k − 1] and sets Si ⊆ Vi such that |Si | = pn/k for each i ∈ [r] there are at least βnp+1 edges in Jp+1 [S], where S := i∈[r] Si .

J copies of ej for each j ∈ [s]. Then χ(T ) − χ(T ) = tq! x. -fold (u, v)-transferral in (J, M ) with size at most tq! ≤ C. Since U ∈ P and u, v ∈ U were arbitrary, we deduce that (J, M ) is (B, C)-irreducible with respect to P, as required. 2. Transferral digraphs To obtain perfect matchings, we need more precise balancing operations, namely simple transferrals, by which we mean 1-fold transferrals. We represent these using digraphs (directed graphs), for which we make the following standard deﬁnitions.

Uk are clusters of P such that U := U1 ∪ · · · ∪ Uk satisﬁes |Z[U ]| ≤ νnk1 and |G[U ]| ≥ dnk1 . Let P denote the partition of U into parts U1 , . . , 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 , .