By Peter Keevash
ISBN-10: 1470409658
ISBN-13: 9781470409654
The authors strengthen a conception for the lifestyles of ideal matchings in hypergraphs below rather basic stipulations. Informally talking, the obstructions to excellent matchings are geometric, and are of 2 targeted kinds: 'space obstacles' from convex geometry, and 'divisibility boundaries' from mathematics lattice-based buildings. To formulate specified effects, they introduce the surroundings of simplicial complexes with minimal measure sequences, that is a generalisation of the standard minimal measure situation. They confirm the primarily absolute best minimal measure series for locating a nearly excellent matching. moreover, their major consequence establishes the soundness estate: less than a similar measure assumption, if there's no excellent matching then there needs to be an area or divisibility barrier. this permits using the soundness procedure in proving particular effects. along with convalescing earlier 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 consequence for Fischer's conjecture; because the specified consequence 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
Example text
In the setting of the theorem, we say that G and H are ν-close (with respect to Q) if |GA HA | < ν|KA (V )| for every A ∈ [r] k . 2. r|n. Let V be a set of n vertices, Q be an balanced partition of V into r parts, and H be a Q-partite k-graph on V . Then there is an a-bounded ε-regular vertex-equitable Q-partition (k − 1)-complex P on V , and an Q-partite k-graph G on V , such that G is ν-close to H and perfectly ε-regular with respect to P. 3. The hypergraph blowup lemma While hypergraph regularity theory is a relatively recent development, still more recent is the hypergraph blow-up lemma due to Keevash [22], which makes it possible to apply hypergraph regularity theory to embeddings of spanning subcomplexes.
Xs ∈ X such that s ≤ n + 1, each λj is q-rational, j∈[s] λj = 1, and x = j∈[s] λj xj . By definition of X, we can choose ej ∈ J and ej ∈ M such that χ(ej ) − χ(ej ) = xj for j ∈ [s]. λj 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.
Vt ∈ V (D) for some t. Let P be the partition of V (D) into parts V1 , . . , Vt . So each part of P has size at least δ + (D) − αn. Now, for any j ∈ [t] and any u ∈ Vj there is a path in D from u to vj of length at most , and therefore an edge in D from u to vj . Then S := ({v1 }, . . , {vt }) is a dominated P-tuple in D , as required. Now we show that in combination with irreducibility, a dominated partition P has the property that some transferral digraph is complete on every part of P. We need the following lemma.
A geometric theory for hypergraph matching by Peter Keevash
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