We show that several problems that are hard for various parameterized complexity classes on general graphs, become fixed parameter tractable on graphs with no small cycles. More specifically, we give fixed parameter algorithms for Dominating Set, t-Vertex Cover (where we need to cover at least t edges) and several of their variants on graphs that have no triangles or cycles of length 4. These p...

We prove there is a countable dense homogeneous subspace of R of size א1. The proof involves an absoluteness argument using an extension of the Lω1ω(Q) logic obtained by adding predicates for Borel sets. A separable topological space X is countable dense homogeneous (CDH) if, given any two countable dense subsets D and D′ of X, there is a homeomorphism h of X such that h[D] = D′. The main purpo...

We analyze the simple greedy algorithm that iteratively removes the endpoints of a maximum-degree edge in a graph, where the degree of an edge is the sum of the degrees of its endpoints. This algorithm provides a 2-approximation to the minimum edge dominating set and minimum maximal matching problems. We refine its analysis and give an expression of the approximation ratio that is strictly less...

We prove that it is relatively consistent with ZFC that in any perfect Polish space, for every nonmeager set A there exists a nowhere dense Cantor set C such that A∩C is nonmeager in C. We also examine variants of this result and establish a measure theoretic analog.