20 research outputs found
Edge-ordered Ramsey numbers
We introduce and study a variant of Ramsey numbers for edge-ordered graphs,
that is, graphs with linearly ordered sets of edges. The edge-ordered Ramsey
number of an edge-ordered graph
is the minimum positive integer such that there exists an edge-ordered
complete graph on vertices such that every 2-coloring of
the edges of contains a monochromatic copy of
as an edge-ordered subgraph of .
We prove that the edge-ordered Ramsey number
is finite for every edge-ordered graph and we obtain better
estimates for special classes of edge-ordered graphs. In particular, we prove
for every bipartite
edge-ordered graph on vertices. We also introduce a natural
class of edge-orderings, called lexicographic edge-orderings, for which we can
prove much better upper bounds on the corresponding edge-ordered Ramsey
numbers.Comment: Minor revision, 16 pages, 1 figure. An extended abstract of this
paper will appeared in the Eurocomb 2019 proceedings in Acta Mathematica
Universitatis Comenianae. The paper has been accepted to the European Journal
of Combinatoric
A hierarchy of randomness for graphs
AbstractIn this paper we formulate four families of problems with which we aim at distinguishing different levels of randomness.The first one is completely non-random, being the ordinary Ramsey–Turán problem and in the subsequent three problems we formulate some randomized variations of it. As we will show, these four levels form a hierarchy. In a continuation of this paper we shall prove some further theorems and discuss some further, related problems
Extremal problems in graphs
In the first part of this thesis we will consider degree sequence results for graphs. An important result of Komlós [39] yields the asymptotically exact minimum degree threshold that ensures a graph contains an -tiling covering an -proportion of the vertices of (for any fixed (0, 1) and graph ). In Chapter 2, we give a degree sequence strengthening of this result. A fundamental result of Kühn and Osthus [46] determines up to an additive constant the minimum degree threshold that forces a graph to contain a perfect -tiling. In Chapter 3, we prove a degree sequence version of this result.
We close this thesis in the study of asymmetric Ramsey properties in . Specifically, for fixed graphs we study the asymptotic threshold function for the property → . Rödl and Ruciński [61, 62, 63] determined the threshold function for the general symmetric case; that is, when . Kohayakawa and Kreuter [33] conjectured the threshold function for the asymmetric case. Building on work of Marciniszyn, Skokan, Spöhel and Steger [51], in Chapter 4, we reduce the 0-statement of Kohayakawa and Kreuter’s conjecture to a more approachable, deterministic conjecture. To demonstrate the potential of this approach, we show our conjecture holds for almost all pairs of regular graphs (satisfying certain balancedness conditions)
Canonical Forms of Borel Functions on the Milliken Space
The goal of this work is to canonize arbitrary Borel measurable mappings on the Milliken space, which is the space of all increasing infinite sequences of pairwise disjoint nonempty finite subsets of the set of all nonnegative integers. The main result refers to the metric topology on the Milliken space. The result is a common generalization of a theorem of Taylor and a theorem of Prömel and Voigt