Mathematics – Logic
Scientific paper
2002-05-31
Mathematics
Logic
29 pages. LaTeX2e
Scientific paper
We investigate the Ramsey theory of continuous pair-colorings on complete, separable metric spaces, and apply the results to the problem of covering a plane by functions. The homogeneity number hm(c) of a pair-coloring c:[X]^2 -> 2 is the number of c-homogeneous subsets of X needed to cover X. We isolate two continuous pair-colorings on the Cantor space 2^omega, c_min and c_max, which satisfy hm(c_min)\le hm(c_max) and prove: 1. For every Polish space X and every continuous pair-coloring c:[X]^2 -> 2 with hm(c) uncountable: hm(c)= hm(c_min) or hm(c)=hm(c_max) 2. There is a model of set theory in which hm(c_min)=aleph_1 and hm(c_max)=aleph_2 (The consistency of hm(c_min) = 2^aleph0 and of hm(c_max) < 2^aleph0 is known) We prove that hm(c_min) is equal to the covering number of (2^omega)^2 by graphs of Lipschitz functions and their reflections on the diagonal. An iteration of an optimal forcing notion associated to c_min gives: There is a model of set theory in which 1. R^2 is coverable by aleph1 graphs and reflections of graphs of continuous real functions; 2. R^2 is not coverable by aleph1 graphs and reflections of graphs of Lipschitz real functions.
Geschke Stefan
Goldstern Martin
Kojman Menachem
No associations
LandOfFree
Continuous Ramsey theory on Polish spaces and covering the plane by functions does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Continuous Ramsey theory on Polish spaces and covering the plane by functions, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Continuous Ramsey theory on Polish spaces and covering the plane by functions will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-75894