Stefan Geschke

Convex decompositions in the plane and continuous pair colorings of the irrationals

AbstractWe address the structure of nonconvex closed subsets of the Euclidean plane. A closed subsetS⊆ℝ2 which is not presentable as a countable union of convex sets satisfies the following dichotomy:(1)There is a perfect nonemptyP⊆S so that |C∩P|<3 for every convexC⊆S. In this case coveringS by convex subsets ofS is equivalent to coveringP by finite subsets, hence no nontrivial convex covers ofS can exist.(2)There exists a continuous pair coloringf: [N]2→{0, 1} of the spaceN of irrational numbers so that coveringS by convex subsets is equivalent to coveringN byf-monochromatic sets. In this case it is consistent thatS has a convex cover of cardinality strictly smaller than the continuumc in some forcing extension of the universe. We also show that iff: [N]2→{0, 1} is a continuous coloring of pairs, and no open subset ofN isf-monochromatic, then the least numberκ off-monochromatic sets required to coverN satisfiesK+>-c. Consequently, a closed subset of ℝ2 that cannot be covered by countably many convex subsets, cannot be covered by any number of convex subsets other than the continuum or the immediate predecessor of the continuum. The analogous fact is false for closed subsets of ℝ3.

Convexity numbers of closed sets in R^n

For n > 2 let I n be the σ-ideal in P(n ω ) generated by all sets which do not contain n equidistant points in the usual metric on n ω . For each n > 2 a set S n is constructed in R n so that the σ-ideal which is generated by the convex subsets of S n restricted to the convexity radical K(S n ) is isomorphic to I n . Thus cov(I n ) is equal to the least number of convex subsets required to cover S n - the convexity number of S n . For every non-increasing function f: ω \ 2 → {κ ∈ card: cf(κ) > N 0 } we construct a model of set theory in which cov(I n ) = f(n) for each n ∈ ω \ 2. When f is strictly decreasing up to n, n - 1 uncountable cardinals are simultaneously realized as convexity numbers of closed subsets of R n . It is conjectured that n, but never more than n, different uncountable cardinals can occur simultaneously as convexity numbers of closed subsets of R n . This conjecture is true for n = 1 and n = 2.

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

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:

On the weak Freese–Nation property of ?(ω)

Abstract. Continuing [6], [8] and [16], we study the consequences of the weak Freese-Nation property of (?(ω),⊆). Under this assumption, we prove that most of the known cardinal invariants including all of those appearing in Cichońs diagram take the same value as in the corresponding Cohen model. Using this principle we could also strengthen two results of W. Just about cardinal sequences of superatomic Boolean algebras in a Cohen model. These results show that the weak Freese-Nation property of (?(ω),⊆) captures many of the features of Cohen models and hence may be considered as a principle axiomatizing a good portion of the combinatorics available in Cohen models.

On the weak Freese-Nation property of complete Boolean algebras

Abstract The following results are proved: (a) In a model obtained by adding ℵ 2 Cohen reals, there is always a c.c.c. complete Boolean algebra without the weak Freese-Nation property. (b) Modulo the consistency strength of a supercompact cardinal, the existence of a c.c.c. complete Boolean algebra without the weak Freese-Nation property is consistent with GCH. (c) If a weak form of □μ and cof ([μ] ℵ 0 ,⊆)=μ + hold for each μ>cf(μ)=ω, then the weak Freese-Nation property of 〈 P (ω),⊆〉 is equivalent to the weak Freese-Nation property of any of C (κ) or R (κ) for uncountable κ. (d) Modulo the consistency of (ℵ ω+1 ,ℵ ω )↠(ℵ 1 ,ℵ 0 ) , it is consistent with GCH that C (ℵ ω ) does not have the weak Freese-Nation property and hence the assertion in (c) does not hold, and also that adding ℵ ω Cohen reals destroys the weak Freese-Nation property of 〈 P (ω), ⊆ 〉 . These results solve all of the problems except Problem 1 in S. Fuchino, L. Soukup, Fundament. Math. 154 (1997) 159–176, and some other problems posed by Geschke.

Covering R^n^+^1 by graphs of n-ary functions and long linear orderings of Turing degrees

A point (x 0 ,...,x n ) ∈ X n+1 is covered by a function f: X n → X iff there is a permutation σ of n + 1 such that x σ(0) = f(x σ(1) ,...,x σ(n) ). By a theorem of Kuratowski, for every infinite cardinal κ exactly κ n-ary functions are needed to cover all of (K +n ) n+1 . We show that for arbitrarily large uncountable κ it is consistent that the size of the continuum is C +n and R n+1 is covered by κ n-ary continuous functions. We study other cardinal invariants of the σ-ideal on R n+1 generated by continuous n-ary functions and finally relate the question of how many continuous functions are necessary to cover R 2 to the least size of a set of parameters such that the Turing degrees relative to this set of parameters are linearly ordered.

Weak Borel chromatic numbers

Given a graph G whose set of vertices is a Polish space X, the weak Borel chromatic number of G is the least size of a family of pairwise disjoint G -independent Borel sets that covers all of X. Here a set of vertices of a graph G is independent if no two vertices in the set are connected by an edge. We show that it is consistent with an arbitrarily large size of the continuum that every closed graph on a Polish space either has a perfect clique or has a weak Borel chromatic number of at most ℵ1. We observe that some weak version of Todorcevics Open Coloring Axiom for closed colorings follows from MA. Slightly weaker results hold for Fσ-graphs. In particular, it is consistent with an arbitrarily large size of the continuum that every locally countable Fσ-graph has a Borel chromatic number of at most ℵ1. We refute various reasonable generalizations of these results to hypergraphs (© 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

More on convexity numbers of closed sets in ℝⁿ

The convexity number of a set S C R n is the least size of a family F of convex sets with ∪F = S. S is countably convex if its convexity number is countable. Otherwise S is uncountably convex. Uncountably convex closed sets in R n have been studied recently by Geschke, Kubis, Kojman and Schipperus. Their line of research is continued in the present article. We show that for all n > 2, it is consistent that there is an uncountably convex closed set S C R n+1 whose convexity number is strictly smaller than all convexity numbers of uncountably convex subsets of R n . Moreover, we construct a closed set S ⊆ R 3 whose convexity number is 2 N 0 and that has no uncountable k-clique for any k > 1. Here C ⊆ C S is a k-clique if the convex hull of no k-element subset of C is included in S. Our example shows that the main result of the above-named authors, a closed set S ⊆ R 2 either has a perfect 3-clique or the convexity number of S is < 2 N 0 in some forcing extension of the universe, cannot be extended to higher dimensions.

APPLICATIONS OF ELEMENTARY SUBMODELS IN GENERAL TOPOLOGY

Elementary submodels of some initial segment of the set-theoretic universe are useful in order to prove certain theorems in general topology as well as in algebra. As an illustration we give proofs of two theorems due to Arkhangel’skii concerning cardinal invariants of compact spaces.

Symmetrized Induced Ramsey Theory

We prove induced Ramsey theorems in which the monochromatic induced subgraph satisfies that all members of a prescribed set of its partial isomorphisms extend to automorphisms of the colored graph (without requirement of preservation of colors). We consider vertex and edge colorings, and extensions of partial isomorphisms in the set of all partial isomorphisms between singletons as considered by Babai and Sós (European J Combin 6(2):101–114, 1985), the set of all finite partial isomorphisms as considered by Hrushovski (Combinatorica 12(4):411–416, 1992), Herwig (Combinatorica 15:365–371, 1995) and Herwig-Lascar (Trans Amer Math Soc 5:1985–2021, 2000), and the set of all total isomorphisms. We observe that every finite graph embeds into a finite vertex transitive graph by a so called bi-embedding, an embedding that is compatible with a monomorphism between the corresponding automorphism groups. We also show that every countable graph bi-embeds into Rado’s universal countable graph Γ.