1,540 research outputs found
On globally sparse Ramsey graphs
We say that a graph has the Ramsey property w.r.t.\ some graph and
some integer , or is -Ramsey for short, if any -coloring
of the edges of contains a monochromatic copy of . R{\"o}dl and
Ruci{\'n}ski asked how globally sparse -Ramsey graphs can possibly
be, where the density of is measured by the subgraph with
the highest average degree. So far, this so-called Ramsey density is known only
for cliques and some trivial graphs . In this work we determine the Ramsey
density up to some small error terms for several cases when is a complete
bipartite graph, a cycle or a path, and colors are available
Embedding large subgraphs into dense graphs
What conditions ensure that a graph G contains some given spanning subgraph
H? The most famous examples of results of this kind are probably Dirac's
theorem on Hamilton cycles and Tutte's theorem on perfect matchings. Perfect
matchings are generalized by perfect F-packings, where instead of covering all
the vertices of G by disjoint edges, we want to cover G by disjoint copies of a
(small) graph F. It is unlikely that there is a characterization of all graphs
G which contain a perfect F-packing, so as in the case of Dirac's theorem it
makes sense to study conditions on the minimum degree of G which guarantee a
perfect F-packing.
The Regularity lemma of Szemeredi and the Blow-up lemma of Komlos, Sarkozy
and Szemeredi have proved to be powerful tools in attacking such problems and
quite recently, several long-standing problems and conjectures in the area have
been solved using these. In this survey, we give an outline of recent progress
(with our main emphasis on F-packings, Hamiltonicity problems and tree
embeddings) and describe some of the methods involved
Combinatorial theorems relative to a random set
We describe recent advances in the study of random analogues of combinatorial
theorems.Comment: 26 pages. Submitted to Proceedings of the ICM 201
Density theorems for bipartite graphs and related Ramsey-type results
In this paper, we present several density-type theorems which show how to
find a copy of a sparse bipartite graph in a graph of positive density. Our
results imply several new bounds for classical problems in graph Ramsey theory
and improve and generalize earlier results of various researchers. The proofs
combine probabilistic arguments with some combinatorial ideas. In addition,
these techniques can be used to study properties of graphs with a forbidden
induced subgraph, edge intersection patterns in topological graphs, and to
obtain several other Ramsey-type statements
A sharp threshold for random graphs with a monochromatic triangle in every edge coloring
Let be the set of all finite graphs with the Ramsey property that
every coloring of the edges of by two colors yields a monochromatic
triangle. In this paper we establish a sharp threshold for random graphs with
this property. Let be the random graph on vertices with edge
probability . We prove that there exists a function with
, as tends to infinity
Pr[G(n,(1-\eps)\hat c/\sqrt{n}) \in \R ] \to 0 and Pr [ G(n,(1+\eps)\hat
c/\sqrt{n}) \in \R ] \to 1. A crucial tool that is used in the proof and is
of independent interest is a generalization of Szemer\'edi's Regularity Lemma
to a certain hypergraph setting.Comment: 101 pages, Final version - to appear in Memoirs of the A.M.
Problems and memories
I state some open problems coming from joint work with Paul Erd\H{o}sComment: This is a paper form of the talk I gave on July 5, 2013 at the
centennial conference in Budapest to honor Paul Erd\H{o}
The history of degenerate (bipartite) extremal graph problems
This paper is a survey on Extremal Graph Theory, primarily focusing on the
case when one of the excluded graphs is bipartite. On one hand we give an
introduction to this field and also describe many important results, methods,
problems, and constructions.Comment: 97 pages, 11 figures, many problems. This is the preliminary version
of our survey presented in Erdos 100. In this version 2 only a citation was
complete
Lower bounds for Max-Cut in -free graphs via semidefinite programming
For a graph , let denote the size of the maximum cut in . The
problem of estimating as a function of the number of vertices and edges
of has a long history and was extensively studied in the last fifty years.
In this paper we propose an approach, based on semidefinite programming (SDP),
to prove lower bounds on . We use this approach to find large cuts in
graphs with few triangles and in -free graphs.Comment: 21 pages, to be published in LATIN 2020 proceedings, Updated version
is rewritten to include additional results along with corrections to original
argument
Erdos-Hajnal-type theorems in hypergraphs
The Erdos-Hajnal conjecture states that if a graph on n vertices is H-free,
that is, it does not contain an induced copy of a given graph H, then it must
contain either a clique or an independent set of size n^{d(H)}, where d(H) > 0
depends only on the graph H. Except for a few special cases, this conjecture
remains wide open. However, it is known that a H-free graph must contain a
complete or empty bipartite graph with parts of polynomial size. We prove an
analogue of this result for 3-uniform hypergraphs, showing that if a 3-uniform
hypergraph on n vertices is H-free, for any given H, then it must contain a
complete or empty tripartite subgraph with parts of order c(log n)^{1/2 +
d(H)}, where d(H) > 0 depends only on H. This improves on the bound of c(log
n)^{1/2}, which holds in all 3-uniform hypergraphs, and, up to the value of the
constant d(H), is best possible. We also prove that, for k > 3, no analogue of
the standard Erdos-Hajnal conjecture can hold in k-uniform hypergraphs. That
is, there are k-uniform hypergraphs H and sequences of H-free hypergraphs which
do not contain cliques or independent sets of size appreciably larger than one
would normally expect.Comment: 15 page
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