3,469 research outputs found
The random graph
Erd\H{o}s and R\'{e}nyi showed the paradoxical result that there is a unique
(and highly symmetric) countably infinite random graph. This graph, and its
automorphism group, form the subject of the present survey.Comment: Revised chapter for new edition of book "The Mathematics of Paul
Erd\H{o}s
Fraisse Limits, Ramsey Theory, and Topological Dynamics of Automorphism Groups
We study in this paper some connections between the Fraisse theory of
amalgamation classes and ultrahomogeneous structures, Ramsey theory, and
topological dynamics of automorphism groups of countable structures.Comment: 73 pages, LaTeX 2e, to appear in Geom. Funct. Ana
Graph removal lemmas
The graph removal lemma states that any graph on n vertices with o(n^{v(H)})
copies of a fixed graph H may be made H-free by removing o(n^2) edges. Despite
its innocent appearance, this lemma and its extensions have several important
consequences in number theory, discrete geometry, graph theory and computer
science. In this survey we discuss these lemmas, focusing in particular on
recent improvements to their quantitative aspects.Comment: 35 page
Large rainbow cliques in randomly perturbed dense graphs
For two graphs and , write if has the property that every {\sl proper} colouring of its edges
yields a {\sl rainbow} copy of .
We study the thresholds for such so-called {\sl anti-Ramsey} properties in
randomly perturbed dense graphs, which are unions of the form , where is an -vertex graph with edge-density at least
, and is a constant that does not depend on .
Our results in this paper, combined with our results in a companion paper,
determine the threshold for the property for every . In this paper, we
show that for the threshold is ; in fact, our -statement is a supersaturation result. This
turns out to (almost) be the threshold for as well, but for every , the threshold is lower; see our companion paper for more details.
In this paper, we also consider the property , and show that the
threshold for this property is for every ; in particular,
it does not depend on the length of the cycle . It is worth
mentioning that for even cycles, or more generally for any fixed bipartite
graph, no random edges are needed at all.Comment: 21 pages; some typos fixed in the last versio
- β¦