735 research outputs found
A hypergraph blow-up lemma
We obtain a hypergraph generalisation of the graph blow-up lemma proved by
Komlos, Sarkozy and Szemeredi, showing that hypergraphs with sufficient
regularity and no atypical vertices behave as if they were complete for the
purpose of embedding bounded degree hypergraphs.Comment: 102 pages, 1 figure, to appear in Random Structures and Algorithm
Counting in hypergraphs via regularity inheritance
We develop a theory of regularity inheritance in 3-uniform hypergraphs. As a simple consequence we deduce a strengthening of a counting lemma of Frankl and Rödl. We believe that the approach is sufficiently flexible and general to permit extensions of our results in the direction of a hypergraph blow-up lemma
Transversals via regularity
Given graphs all on the same vertex set and a graph with
, a copy of is transversal or rainbow if it contains at most
one edge from each . When , such a copy contains exactly one edge
from each . We study the case when is spanning and explore how the
regularity blow-up method, that has been so successful in the uncoloured
setting, can be used to find transversals. We provide the analogues of the
tools required to apply this method in the transversal setting. Our main result
is a blow-up lemma for transversals that applies to separable bounded degree
graphs .
Our proofs use weak regularity in the -uniform hypergraph whose edges are
those where is an edge in the graph . We apply our lemma to
give a large class of spanning -uniform linear hypergraphs such that any
sufficiently large uniformly dense -vertex -uniform hypergraph with
minimum vertex degree contains as a subhypergraph. This
extends work of Lenz, Mubayi and Mycroft
Strong Jumps and Lagrangians of Non-Uniform Hypergraphs
The hypergraph jump problem and the study of Lagrangians of uniform
hypergraphs are two classical areas of study in the extremal graph theory. In
this paper, we refine the concept of jumps to strong jumps and consider the
analogous problems over non-uniform hypergraphs. Strong jumps have rich
topological and algebraic structures. The non-strong-jump values are precisely
the densities of the hereditary properties, which include the Tur\'an densities
of families of hypergraphs as special cases. Our method uses a generalized
Lagrangian for non-uniform hypergraphs. We also classify all strong jump values
for -hypergraphs.Comment: 19 page
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