Continuous Ramsey Theory on Polish Spaces and Covering the Plane by Functions
نویسندگان
چکیده
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 2ω , cmin and cmax, which satisfy hm(cmin) ≤ hm(cmax) and prove: Theorem. (1) For every Polish space X and every continuous pair-coloring c : [X]2 → 2 with hm(c) > א0, hm(c) = hm(cmin) or hm(c) = hm(cmax). (2) There is a model of set theory in which hm(cmin) = א1 and hm(cmax) = א2. The consistency of hm(cmin) = 2 א0 and of hm(cmax) < 2א0 follows from [20]. We prove that hm(cmin) is equal to the covering number of (2 ω)2 by graphs of Lipschitz functions and their reflections on the diagonal. An iteration of an optimal forcing notion associated to cmin gives: Theorem. There is a model of set theory in which (1) R2 is coverable by א1 graphs and reflections of graphs of continuous real functions; (2) R2 is not coverable by א1 graphs and reflections of graphs of Lipschitz real functions. Diagram 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 Diagram 1 can be separated if one excludes Cov(Lip(R)) from row (3).
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