By Peter Keevash

ISBN-10: 1470409658

ISBN-13: 9781470409654

The authors strengthen a concept for the lifestyles of ideal matchings in hypergraphs below really normal stipulations. Informally talking, the obstructions to excellent matchings are geometric, and are of 2 unique forms: 'space limitations' from convex geometry, and 'divisibility limitations' from mathematics lattice-based buildings. To formulate targeted effects, they introduce the environment of simplicial complexes with minimal measure sequences, that's a generalisation of the standard minimal measure . They confirm the primarily very best minimal measure series for locating a virtually excellent matching. in addition, their major outcome establishes the soundness estate: less than an analogous measure assumption, if there is not any ideal matching then there has to be an area or divisibility barrier. this permits using the soundness technique in proving specified effects. along with getting better prior effects, the authors practice our thought 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 turn out the precise outcome for tetrahedra and the asymptotic end result for Fischer's conjecture; because the precise outcome for the latter is technical they defer it to a next paper

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**Additional info for A geometric theory for hypergraph matching**

**Example text**

Now consider S = i∈[r],p

4, the output G \ Z is a subgraph of H. 4. (Blow-up Lemma) Suppose 1/n ε d∗ d, c, 1/k, 1/D, 1/C. Let V be a set of vertices, Q be a partition of V into k parts V1 , . . , Vk with n ≤ |Vj | ≤ Cn for each j ∈ [k], and G be an ε-regular Q-partite k-complex on V such that |G{j} | = |Vj | for each j ∈ [k], d[k] (G) ≥ d and d(G) ≥ da . Suppose Z ⊆ Gk satisﬁes |Z| ≤ θ|Gk |. Then we can delete at most 2θ 1/3 |Vj | vertices from each Vj to obtain V = V1 ∪ · · · ∪ Vk , G = G[V ] and Z = Z[V ] such that (i) d(G ) > d∗ and |G (v)k | > d∗ |Gk |/|Vi | for every v ∈ Vi , and (ii) G \ Z is c-robustly D-universal.

To prove the claim, we start by showing that there must be some v ∈ S with (v) := |N − (v) ∩ S| ≥ δ + (D) − αn. To see this, we note that v∈S d− d− S S (v) ≥ + + 2 d (v) − L(S) ≥ |S|δ (D) − γn . Thus by averaging we can choose v ∈ v∈S − + 2 + S with dS (v) ≥ δ (D) − γn /|S| ≥ δ (D) − αn. Now consider the ‘iterated inneighbourhood’ Nj− of v for j ≥ 1, deﬁned as the set of vertices u ∈ S such that there exists a path from u to v in D of length at most j. Note that N1− ⊆ N2− ⊆ . . , − | ≤ |Nj− |+γn.

### A geometric theory for hypergraph matching by Peter Keevash

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