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Perfect Packings in Quasirandom Hypergraphs II

Part of: Graph theory Extremal combinatorics

Published online by Cambridge University Press:  27 October 2015

JOHN LENZ  and
DHRUV MUBAYI
JOHN LENZ
Affiliation:
Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, IL 60607, USA (e-mail: [email protected], [email protected])
DHRUV MUBAYI
Affiliation:
Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, IL 60607, USA (e-mail: [email protected], [email protected])
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Abstract

For each of the notions of hypergraph quasirandomness that have been studied, we identify a large class of hypergraphs F so that every quasirandom hypergraph H admits a perfect F-packing. An informal statement of a special case of our general result for 3-uniform hypergraphs is as follows. Fix an integer r ⩾ 4 and 0 < p < 1. Suppose that H is an n-vertex triple system with r|n and the following two properties:

  • • for every graph G with V(G) = V(H), at least p proportion of the triangles in G are also edges of H,

  • • for every vertex x of H, the link graph of x is a quasirandom graph with density at least p.

Then H has a perfect Kr(3)-packing. Moreover, we show that neither of the hypotheses above can be weakened, so in this sense our result is tight. A similar conclusion for this special case can be proved by Keevash's Hypergraph Blow-up Lemma, with a slightly stronger hypothesis on H.

MSC classification

Primary: 05C65: Hypergraphs
Secondary: 05C70: Factorization, matching, partitioning, covering and packing 05D40: Probabilistic methods
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 25 , Issue 4 , July 2016 , pp. 595 - 611
Copyright
Copyright © Cambridge University Press 2015 

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