Combinatorics

Download A geometric theory for hypergraph matching by Peter Keevash PDF

By Peter Keevash

The authors boost a concept for the lifestyles of ideal matchings in hypergraphs lower than really common stipulations. Informally talking, the obstructions to ideal matchings are geometric, and are of 2 precise varieties: 'space limitations' from convex geometry, and 'divisibility obstacles' from mathematics lattice-based buildings. To formulate specified effects, they introduce the surroundings of simplicial complexes with minimal measure sequences, that's a generalisation of the standard minimal measure . They verify the primarily absolute best minimal measure series for locating a virtually ideal matching. moreover, their major end result establishes the soundness estate: lower than an analogous measure assumption, if there is not any excellent matching then there needs to be an area or divisibility barrier. this permits using the steadiness approach in proving unique effects. in addition to improving prior effects, the authors practice 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 end result for tetrahedra and the asymptotic outcome for Fischer's conjecture; because the designated outcome for the latter is technical they defer it to a next paper

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Irreducibility Now we make some important definitions. Given b ∈ N, u, v ∈ V (J) and multisets T in J and T in M , we say that (T, T ) is a b-fold (u, v)-transferral in (J, M ) if χ(T ) − χ(T ) = b(χ({u}) − χ({v})). That is to say that every vertex of J appears equally many times in T as in T , with the exception of u, which appears b times more in T than in T , and v, which appears b times more in T than in T . An example is shown in Figure 3. Note that if (T, T ) is a b-fold (u, v)-transferral in (J, M ) then we must have |T | = |T |; we refer to this common size as the size of the transferral.

Xr respectively. Fix any f ∈ F , j ∈ [k − 1] and S = {u1 , . . , uj } ∈ Rj with ui ∈ Uf (i) for i ∈ [j]. Then S is QR -partite and PR F -partite, and |J[VS ]| ≥ cj nj1 . For any edge e ∈ J[VS ], we may write e = {v1 , . . , vj } with vi ∈ Xf (i) for i ∈ [j]. There are at least δjF (J) vertices vj+1 ∈ Xf (j+1) such that {v1 , . . , vj+1 } ∈ J, and of these at most jrn/h belong to the same part of Q as one of v1 , . . , vj . Thus we obtain at least |J[VS ]|(δjF (J) − jrn/h) edges in sets J[VT ], T ∈ T , where T denotes the collection of QR -partite sets T = S ∪ {u} for some u ∈ Uf (j+1) \ S.

Then there is some i ∈ [r] such that a is not constant on Vi . We label V so that for each i ∈ [r], the vertices of Vi are labelled vi,1 , . . , vi,n , the corresponding coordinates of a are ai,1 , . . , ai,n , and ai,1 ≤ ai,2 ≤ · · · ≤ ai,n . It is convenient to assume that we have a strict inequality δiF (J) > k−i k − α n for i ∈ [k − 1]. This can be achieved by replacing α with a slightly smaller value; we can also assume that αn ∈ N. Next we partition each Vi into k parts Vi,1 , . . , Vi,k as follows.

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