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New PDF release: A geometric theory for hypergraph matching

By Peter Keevash

The authors advance a thought for the life of excellent matchings in hypergraphs below particularly basic stipulations. Informally talking, the obstructions to excellent matchings are geometric, and are of 2 distinctive forms: 'space obstacles' from convex geometry, and 'divisibility limitations' from mathematics lattice-based structures. To formulate special effects, they introduce the environment of simplicial complexes with minimal measure sequences, that is a generalisation of the standard minimal measure situation. They confirm the basically very best minimal measure series for locating a nearly ideal matching. moreover, their major end result establishes the steadiness estate: below a similar measure assumption, if there isn't any ideal matching then there needs to be an area or divisibility barrier. this permits using the soundness procedure in proving unique effects. in addition to improving earlier 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. the following they end up the precise consequence for tetrahedra and the asymptotic outcome for Fischer's conjecture; because the unique end result for the latter is technical they defer it to a next paper

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The authors strengthen a concept for the life of excellent matchings in hypergraphs less than really normal stipulations. Informally conversing, the obstructions to ideal matchings are geometric, and are of 2 precise forms: 'space boundaries' from convex geometry, and 'divisibility limitations' from mathematics lattice-based buildings.

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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. Indeed, it is similar to the blow-up lemma for graphs, insomuch as it states that by deleting a small number of vertices from a regular k-complex we may obtain a super-regular complex, in which we can find any spanning subcomplex of bounded maximum degree.

An allocation function is a function f : [k] → [r]. Intuitively an allocation function should be viewed as a rule for forming an edge of J by choosing vertices one by one; the ith vertex should be chosen from part Vf (i) of P. This naturally leads to the notion of the minimum f -degree sequence f (J) , δ f (J) := δ0f (J), . . , δk−1 where δjf (J) is the largest m such that for any {v1 , . . , vj } ∈ J with vi ∈ Vf (i) for i ∈ [j] there are at least m vertices vj+1 ∈ Vf (j+1) such that {v1 , .

By irreducibility, for each i ∈ [k|S|] there is a b-fold (ui , vi )-transferral in (J, M ) of size c for some b ≤ B and c ≤ C. -fold (ui , vi )-transferral (Ti , Ti ) of size at most CB!. S + i∈[k|S|] Ti . Then |T | = |T | ≤ CB! + 2Ck · CB! ≤ C . (χ(S) − χ(S )) + χ(Ti ) − χ(Ti ) i∈[k|S|] χ({ui }) − χ({vi }) − χ({ui }) + χ({vi }) = 0, = B! i∈[k|S|] which completes the proof. Next we need a simple proposition that allows us to efficiently represent vectors in a lattice, in that we have bounds on the number of terms and size of the coefficients in the representation.

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