21 research outputs found
The multicolour size-Ramsey number of powers of paths
Given a positive integer s, a graph G is s-Ramsey for a graph H, denoted G→(H)s, if every s-colouring of the edges of G contains a monochromatic copy of H. The s-colour size-Ramsey number rˆs(H) of a graph H is defined to be rˆs(H)=min{|E(G)|:G→(H)s}. We prove that, for all positive integers k and s, we have rˆs(Pnk)=O(n), where Pnk is the kth power of the n-vertex path Pn
On the structure of graphs with forbidden induced substructures
One of the central goals in extremal combinatorics is to understand how the global structure of a combinatorial object, e.g. a graph, hypergraph or set system, is affected by local constraints.
In this thesis we are concerned with structural properties of graphs and hypergraphs which locally do not look like some type of forbidden induced pattern. Patterns can be single subgraphs, families of subgraphs, or in the multicolour version colourings or families of colourings of subgraphs.
Erdős and Szekeres\u27s quantitative version of Ramsey\u27s theorem asserts that in every -edge-colouring of the complete graph on vertices there is a monochromatic clique on at least vertices. The famous Erdős-Hajnal conjecture asserts that forbidding fixed colourings on subgraphs ensures much larger monochromatic cliques. The conjecture is open in general, though a few partial results are known. The first part of this thesis will be concerned with different variants of this conjecture: A bipartite variant, a multicolour variant, and an order-size variant for hypergraphs.
In the second part of this thesis we focus more on order-size pairs; an order-size pair is the family consisting of all graphs of order and size , i.e. on vertices with edges. We consider order-size pairs in different settings: The graph setting, the bipartite setting and the hypergraph setting. In all these settings we investigate the existence of absolutely avoidable pairs, i.e. fixed pairs that are avoided by all order-size pairs with sufficiently large order, and also forcing densities of order-size pairs , i.e. for approaching infinity, the limit superior of the fraction of all possible sizes , such that the order-size pair does not avoid the pair
Graphs without a rainbow path of length 3
In 1959 Erd\H{o}s and Gallai proved the asymptotically optimal bound for the
maximum number of edges in graphs not containing a path of a fixed length. Here
we study a rainbow version of their theorem, in which one considers
graphs on a common set of vertices not creating a path having edges from
different graphs and asks for the maximal number of edges in each graph. We
prove the asymptotically optimal bound in the case of a path on three edges and
any
On multicolor Ramsey numbers of triple system paths of length 3
Let be a 3-uniform hypergraph. The multicolor Ramsey number is the smallest integer such that every coloring of with colors has a monochromatic copy of . Let
be the loose 3-uniform path with 3 edges and
denote the messy 3-uniform path with 3 edges; that is, let and . In this note we
prove and for
sufficiently large. The former result improves on the bound , which was recently established by {\L}uczak and Polcyn.Comment: 18 pages, 3 figure