Continuous Ramsey theory on polish spaces and covering the plane by functions

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

Forcing with copies of the Rado and Henson graphs.Osvaldo Guzmán & Stevo Todorcevic - 2023 - Annals of Pure and Applied Logic 174 (8):103286.
Hypergraphs and proper forcing.Jindřich Zapletal - 2019 - Journal of Mathematical Logic 19 (2):1950007.
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.
Potential continuity of colorings.Stefan Geschke - 2008 - Archive for Mathematical Logic 47 (6):567-578.

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