## Continuous Ramsey theory on Polish spaces and covering the plane by functions

Citations: | 1 - 0 self |

### BibTeX

@MISC{Geschke_continuousramsey,

author = {Stefan Geschke and Martin Goldstern and Menachem Kojman},

title = {Continuous Ramsey theory on Polish spaces and covering the plane by functions},

year = {}

}

### OpenURL

### Abstract

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 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 ω, 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) =