1,076 research outputs found
Quasi-Random Influences of Boolean Functions
We examine a hierarchy of equivalence classes of quasi-random properties of
Boolean Functions. In particular, we prove an equivalence between a number of
properties including balanced influences, spectral discrepancy, local strong
regularity, homomorphism enumerations of colored or weighted graphs and
hypergraphs associated with Boolean functions as well as the th-order strict
avalanche criterion amongst others. We further construct families of
quasi-random boolean functions which exhibit the properties of our equivalence
theorem and separate the levels of our hierarchy.Comment: 27 pages, 6 figure
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
Embeddings and Ramsey numbers of sparse k-uniform hypergraphs
Chvatal, Roedl, Szemeredi and Trotter proved that the Ramsey numbers of
graphs of bounded maximum degree are linear in their order. In previous work,
we proved the same result for 3-uniform hypergraphs. Here we extend this result
to k-uniform hypergraphs, for any integer k > 3. As in the 3-uniform case, the
main new tool which we prove and use is an embedding lemma for k-uniform
hypergraphs of bounded maximum degree into suitable k-uniform `quasi-random'
hypergraphs.Comment: 24 pages, 2 figures. To appear in Combinatoric
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
Hamilton cycles in graphs and hypergraphs: an extremal perspective
As one of the most fundamental and well-known NP-complete problems, the
Hamilton cycle problem has been the subject of intensive research. Recent
developments in the area have highlighted the crucial role played by the
notions of expansion and quasi-randomness. These concepts and other recent
techniques have led to the solution of several long-standing problems in the
area. New aspects have also emerged, such as resilience, robustness and the
study of Hamilton cycles in hypergraphs. We survey these developments and
highlight open problems, with an emphasis on extremal and probabilistic
approaches.Comment: to appear in the Proceedings of the ICM 2014; due to given page
limits, this final version is slightly shorter than the previous arxiv
versio
Tournaments, 4-uniform hypergraphs, and an exact extremal result
We consider -uniform hypergraphs with the maximum number of hyperedges
subject to the condition that every set of vertices spans either or
exactly hyperedges and give a construction, using quadratic residues, for
an infinite family of such hypergraphs with the maximum number of hyperedges.
Baber has previously given an asymptotically best-possible result using random
tournaments. We give a connection between Baber's result and our construction
via Paley tournaments and investigate a `switching' operation on tournaments
that preserves hypergraphs arising from this construction.Comment: 23 pages, 6 figure
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