854 research outputs found

    Davies-trees in infinite combinatorics

    Full text link
    This short note, prepared for the Logic Colloquium 2014, provides an introduction to Davies-trees and presents new applications in infinite combinatorics. In particular, we give new and simple proofs to the following theorems of P. Komj\'ath: every nn-almost disjoint family of sets is essentially disjoint for any n∈Nn\in \mathbb N; R2\mathbb R^2 is the union of n+2n+2 clouds if the continuum is at most ℵn\aleph_n for any n∈Nn\in \mathbb N; every uncountably chromatic graph contains nn-connected uncountably chromatic subgraphs for every n∈Nn\in \mathbb N.Comment: 8 pages, prepared for the Logic Colloquium 201

    Separable reduction theorems by the method of elementary submodels

    Get PDF
    We introduce an interesting method of proving separable reduction theorems - the method of elementary submodels. We are studying whether it is true that a set (function) has given property if and only if it has this property with respect to a special separable subspace, dependent only on the given set (function). We are interested in properties of sets "to be dense, nowhere dense, meager, residual or porous" and in properties of functions "to be continuous, semicontinuous or Fr\'echet differentiable". Our method of creating separable subspaces enables us to combine our results, so we easily get separable reductions of function properties such as "be continuous on a dense subset", "be Fr\'echet differentiable on a residual subset", etc. Finally, we show some applications of presented separable reduction theorems and demonstrate that some results of Zajicek, Lindenstrauss and Preiss hold in nonseparable setting as well.Comment: 27 page

    Elementary submodels in infinite combinatorics

    Get PDF
    The usage of elementary submodels is a simple but powerful method to prove theorems, or to simplify proofs in infinite combinatorics. First we introduce all the necessary concepts of logic, then we prove classical theorems using elementary submodels. We also present a new proof of Nash-Williams's theorem on cycle-decomposition of graphs, and finally we improve a decomposition theorem of Laviolette concerning bond-faithful decompositions of graphs
    • …
    corecore