By Peter Keevash

The authors advance a concept for the lifestyles of excellent matchings in hypergraphs below fairly basic stipulations. Informally talking, the obstructions to ideal matchings are geometric, and are of 2 particular kinds: 'space obstacles' from convex geometry, and 'divisibility obstacles' from mathematics lattice-based structures. To formulate exact 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 absolute best minimal measure series for locating a virtually excellent matching. additionally, their major outcome establishes the soundness estate: lower than an identical measure assumption, if there is not any excellent matching then there needs to be an area or divisibility barrier. this permits using the steadiness strategy in proving designated effects. in addition to improving past effects, the authors practice our concept 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 turn out the precise outcome for tetrahedra and the asymptotic end result for Fischer's conjecture; because the special consequence for the latter is technical they defer it to a next paper

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Indeed, if (S, S ) is a simple (u, v)-transferral of size at most and (T, T ) is a simple (v, w)-transferral of size at most , then (S + T, S + T ) is a simple (u, w)-transferral of size at most + . In the last property we used the notation A + B for the multiset union of multisets A and B. We will also use the notation i∈[t] Ai for the multiset union of multisets Ai , i ∈ [t], and the notation pA for the multiset union of p copies of A. 8, for example as shown in Figures 1 and 2). 3, where |S| = 2n/3, and M is a perfect matching in J, then the matched 3-complex (J, M ) does not contain a b-fold (u, v)-transferral for any b ∈ N, u ∈ S and v ∈ / S.

Vk } ∈ Jk in which vj ∈ Vk−j+1 attains at least the maximum value uk−j+1 of a on Vk−j+1 . We use this fact repeatedly in the proof. Let vn ∈ Vk be a vertex of V with maximum a-coordinate; then greedily extending {vn } ∈ J1 to an edge e ∈ Jk as described above gives a · χ(e) ≥ j∈[k] uj =: U . In particular, the definition of a then implies that any edge e ∈ M must have a · χ(e ) ≥ U . However, since M is a perfect matching in Jk we must have e ∈M a · χ(e ) = a · 1 ≈ U |M |. Indeed, the final approximation would be an equality if every Vj had the same size.

Uk , and G be a Q-partite k-complex on U which is c-robustly 2k -universal. Suppose we have Xj ⊆ Uj for each j ∈ [k] such that (i) |Xj | ≥ c|Uj | for each j ∈ [k], and (ii) |Gk [X ∪ {v}](v)| ≥ c|Gk (v)| for any v ∈ U , where X := X1 ∪ · · · ∪ Xk . Then for any sets Wj with Xj ⊆ Wj ⊆ Uj for j ∈ [k] and |W1 | = · · · = |Wk |, writing W = W1 ∪ · · · ∪ Wk , the k-complex G[W ] contains a perfect matching, Proof. By (i) we have |Wj | ≥ |Xj | ≥ c|Uj | for each j, and by (ii) we have |Gk [W ](v)| ≥ |Gk [X ∪ {v}](v)| ≥ c|Gk (v)| for every v ∈ W .

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