5,003 research outputs found
A polynomial regularity lemma for semi-algebraic hypergraphs and its applications in geometry and property testing
Fox, Gromov, Lafforgue, Naor, and Pach proved a regularity lemma for
semi-algebraic -uniform hypergraphs of bounded complexity, showing that for
each the vertex set can be equitably partitioned into a bounded
number of parts (in terms of and the complexity) so that all but an
-fraction of the -tuples of parts are homogeneous. We prove that
the number of parts can be taken to be polynomial in . Our improved
regularity lemma can be applied to geometric problems and to the following
general question on property testing: is it possible to decide, with query
complexity polynomial in the reciprocal of the approximation parameter, whether
a hypergraph has a given hereditary property? We give an affirmative answer for
testing typical hereditary properties for semi-algebraic hypergraphs of bounded
complexity
Tur\'annical hypergraphs
This paper is motivated by the question of how global and dense restriction
sets in results from extremal combinatorics can be replaced by less global and
sparser ones. The result we consider here as an example is Turan's theorem,
which deals with graphs G=([n],E) such that no member of the restriction set
consisting of all r-tuples on [n] induces a copy of K_r.
Firstly, we examine what happens when this restriction set is replaced just
by all r-tuples touching a given m-element set. That is, we determine the
maximal number of edges in an n-vertex such that no K_r hits a given vertex
set.
Secondly, we consider sparse random restriction sets. An r-uniform hypergraph
R on vertex set [n] is called Turannical (respectively epsilon-Turannical), if
for any graph G on [n] with more edges than the Turan number ex(n,K_r)
(respectively (1+\eps)ex(n,K_r), no hyperedge of R induces a copy of K_r in G.
We determine the thresholds for random r-uniform hypergraphs to be Turannical
and to epsilon-Turannical.
Thirdly, we transfer this result to sparse random graphs, using techniques
recently developed by Schacht [Extremal results for random discrete structures]
to prove the Kohayakawa-Luczak-Rodl Conjecture on Turan's theorem in random
graphs.Comment: 33 pages, minor improvements thanks to two referee
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
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