Journal of Mathematical Logic 4 (2):109-145 (2004)

Abstract
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.
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DOI 10.1142/s0219061304000334
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References found in this work BETA

Many Simple Cardinal Invariants.Martin Goldstern & Saharon Shelah - 1993 - Archive for Mathematical Logic 32 (3):203-221.
Monadic Theory of Order and Topology in ZFC.Yuri Gurevich & Saharon Shelah - 1982 - Annals of Mathematical Logic 23 (2-3):179-198.
Chains and Antichains in P(Ω).James E. Baumgartner - 1980 - Journal of Symbolic Logic 45 (1):85-92.

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Citations of this work BETA

Hypergraphs and Proper Forcing.Jindřich Zapletal - 2019 - Journal of Mathematical Logic 19 (2):1950007.
Potential Continuity of Colorings.Stefan Geschke - 2008 - Archive for Mathematical Logic 47 (6):567-578.
A Dual Open Coloring Axiom.Stefan Geschke - 2006 - Annals of Pure and Applied Logic 140 (1):40-51.
Low-Distortion Embeddings of Infinite Metric Spaces Into the Real Line.Stefan Geschke - 2009 - Annals of Pure and Applied Logic 157 (2-3):148-160.

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