Dániel T. Soukup

The Collins–Roscoe mechanism and D-spaces

We prove that if a space X is well ordered (αA), or linearly semi-stratifiable, or elastic then X is a D-space.

Trees, ladders and graphs

We introduce a new method to construct uncountably chromatic graphs from non-special trees and ladder systems. Answering a question of P. Erd?s and A. Hajnal from 1985, we construct graphs of chromatic number ω 1 without uncountable ω-connected subgraphs. Second, we build triangle free graphs of chromatic number ω 1 without subgraphs isomorphic to H ω , ω + 2 .

Uncountable Strongly Surjective Linear Orders

We call a linear order L strongly surjective if whenever K is a suborder of L then there is a surjective f : L → K so that x ≤ y implies f(x) ≤ f(y). We prove various results on the existence and non-existence of uncountable strongly surjective linear orders answering questions of R. Camerlo, R. Carroy and A. Marcone. In particular, ♢+ implies the existence of a lexicographically ordered Suslin tree which is strongly surjective and minimal; every strongly surjective linear order must be an Aronszajn type under 2ℵ0<2ℵ1

INFINITE COMBINATORICS PLAIN AND SIMPLE

2^{\aleph _{0}}<2^{\aleph _{1}}

Partitioning bases of topological spaces

or in the Cohen and other canonical models (where 2ℵ0=2ℵ1

A counterexample in the theory of D-spaces

2^{\aleph _{0}}= 2^{\aleph _{1}}

Decompositions of edge-colored infinite complete graphs into monochromatic paths

); finally, we prove that it is consistent with CH that there are no uncountable strongly surjective linear orders at all.

Properties D and aD are different

We explore a general method based on trees of elementary submodels in order to present highly simplified proofs to numerous results in infinite combinatorics. While countable elementary submodels have been employed in such settings already, we significantly broaden this framework by developing the corresponding technique for countably closed models of size continuum. The applications range from various theorems on paradoxical decompositions of the plane, to coloring sparse set systems, results on graph chromatic number and constructions from point-set topology. Our main purpose is to demonstrate the ease and wide applicability of this method in a form accessible to anyone with a basic background in set theory and logic.

Davies-trees in infinite combinatorics

Abstract We investigate the general question whether a base for a topological space without isolated points can be partitioned into two bases. We prove that every base for a T 3 Lindelof topology can be partitioned into two bases while there exists a consistent example of a first countable, 0-dimensional, Hausdorff space which admits a base without a partition to two bases.