We present some techniques in c.c.c. forcing, and apply them to prove consistency results concerning the isomorphism and embeddability relations on the family of ℵ 1 -dense sets of real numbers. In this direction we continue the work of Baumgartner [2] who proved the axiom BA stating that every two ℵ 1 -dense subsets of R are isomorphic, is consistent. We e.g. prove Con). Let K H , be the set of order types of ℵ 1 -dense homogeneous subsets of (...) R with the relation of embeddability. We prove that for every finite model L , : Con iff L is a distributive lattice. We prove that it is consistent that the Magidor-Malitz language is not countably compact. We deal with the consistency of certain topological partition theorems. E.g. We prove that MA is consistent with the axiom OCA which says: “If X is a second countable space of power ℵ 1 , and { U 0 ,\h.;, U n−1 } is a cover of D ▪ X x X -} x,x> ¦ x ϵ X } consisting of symmetric open sets, then X can be partitioned into { X i \brvbar; i ϵ ω } such that for every i ϵ ω there is l such that D ⊇ U l ”. We also prove that MA+OCA [xrArr] 2 ℵ 0 = ℵ 2. (shrink)

We present a survey of some results and problems concerning constructions which require a diagonalization of length continuum to be carried out, particularly constructions of almost disjoint families of various sorts. We emphasize the role of cardinal invariants of the continuum and their combinatorial characterizations in such constructions.

James Earl Baumgartner (March 23, 1943–December 28, 2011) came of age mathematically during the emergence of forcing as a fundamental technique of set theory, and his seminal research changed the way set theory is done. He made fundamental contributions to the development of forcing, to our understanding of uncountable orders, to the partition calculus, and to large cardinals and their ideals. He promulgated the use of logic such as absoluteness and elementary submodels to solve problems in set theory, he applied (...) his knowledge of set theory to a variety of areas in collaboration with other mathematicians, and he encouraged a community of mathematicians with engaging survey talks, enthusiastic discussions of open problems, and friendly mathematical conversations. (shrink)

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[Formula: see text] 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 2ω, c min and c max, which satisfy [Formula: see text] and prove: Theorem. For every Polish space X and every continuous pair-coloringc:[X]2→2with[Formula: see (...) text], [Formula: see text] There is a model of set theory in which[Formula: see text]and[Formula: see text]. The consistency of [Formula: see text] and of [Formula: see text] follows from [20]. We prove that [Formula: see text] is equal to the covering number of 2 by graphs of Lipschitz functions and their reflections on the diagonal. An iteration of an optimal forcing notion associated to c min gives: Theorem. There is a model of set theory in which ℝ2 is coverable byℵ1graphs and reflections of graphs of continuous real functions; ℝ2 is not coverable byℵ1graphs and reflections of graphs of Lipschitz real functions. Figure 1.1 in the introduction summarizes the ZFC results in Part I of the paper. The independence results in Part II show that any two rows in Fig. 1.1 can be separated if one excludes [Formula: see text] from row. (shrink)