173 research outputs found
Quasirandomness in hypergraphs
An -vertex graph of edge density is considered to be quasirandom
if it shares several important properties with the random graph . A
well-known theorem of Chung, Graham and Wilson states that many such `typical'
properties are asymptotically equivalent and, thus, a graph possessing one
such property automatically satisfies the others.
In recent years, work in this area has focused on uncovering more quasirandom
graph properties and on extending the known results to other discrete
structures. In the context of hypergraphs, however, one may consider several
different notions of quasirandomness. A complete description of these notions
has been provided recently by Towsner, who proved several central equivalences
using an analytic framework. We give short and purely combinatorial proofs of
the main equivalences in Towsner's result.Comment: 19 page
Quasirandom permutations are characterized by 4-point densities
For permutations π and τ of lengths |π|≤|τ| , let t(π,τ) be the probability that the restriction of τ to a random |π| -point set is (order) isomorphic to π . We show that every sequence {τj} of permutations such that |τj|→∞ and t(π,τj)→1/4! for every 4-point permutation π is quasirandom (that is, t(π,τj)→1/|π|! for every π ). This answers a question posed by Graham
The Poset of Hypergraph Quasirandomness
Chung and Graham began the systematic study of k-uniform hypergraph
quasirandom properties soon after the foundational results of Thomason and
Chung-Graham-Wilson on quasirandom graphs. One feature that became apparent in
the early work on k-uniform hypergraph quasirandomness is that properties that
are equivalent for graphs are not equivalent for hypergraphs, and thus
hypergraphs enjoy a variety of inequivalent quasirandom properties. In the past
two decades, there has been an intensive study of these disparate notions of
quasirandomness for hypergraphs, and an open problem that has emerged is to
determine the relationship between them.
Our main result is to determine the poset of implications between these
quasirandom properties. This answers a recent question of Chung and continues a
project begun by Chung and Graham in their first paper on hypergraph
quasirandomness in the early 1990's.Comment: 43 pages, 1 figur
Perfect Packings in Quasirandom Hypergraphs II
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 -packing. Moreover, we show that neither
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
blowup lemma, with a slightly stronger hypothesis on H.Comment: 17 page
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