Martin Goldstern

Continuous Fraïssé Conjecture

We will investigate the relation of countable closed linear orderings with respect to continuous monotone embeddability and will show that there are exactly

CONTINUOUS RAMSEY THEORY ON POLISH SPACES AND COVERING THE PLANE BY FUNCTIONS

\aleph_1

Clones from creatures

many equivalence classes with respect to this embeddability relation. This is an extension of Laver’s result (Laver, Ann. Math. 93(2):89–111, 1971), who considered (plain) embeddability, which yields coarser equivalence classes. Using this result we show that there are only

New reals: Can live with them, can live without them

\aleph_0

The left side of Cichoń’s diagram

many different Gödel logics.

Borel conjecture and dual Borel conjecture

We investigate the Ramsey theory of continuous graph-structures on complete, separable metric spaces and apply the results to the problem of covering a plane by functions. Let the homogeneity number hm(c) of a pair-coloring c : (X) 2 ! 2 be the number of c-homogeneous subsets of X needed to cover X. We isolate two continuous pair-colorings on the Cantor space 2 ! , cmin and cmax, which satisfy hm(cmin) hm(cmax) and prove:

CLONES FROM IDEALS

We show that (consistently) there is a clone C on a countable set such that the interval of clones above C is linearly ordered and has no coatoms.

Interpolation of monotone functions in lattices

We give a self-contained proof of the preservation theorem for proper countable support iterations known as “tools-preservation”, “Case A” or “first preservation theorem” in the literature. We do not assume that the forcings add reals. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Some Classes of Centralizing Monoids on a Three-Element Set

Using a finite support iteration of ccc forcings, we construct a model of

Creature forcing and five cardinal characteristics in Cichoń’s diagram

\aleph_1<\mathrm{add}(\mathcal{N})<\mathrm{cov}(\mathcal{N})<\mathfrak{b}<\mathrm{non}(\mathcal{M})<\mathrm{cov}(\mathcal{M})=\mathfrak{c}