Continuous Ramsey theory on Polish spaces and covering the plane by   functions
  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. 
Stefan Geschke (FU Berlin), Martin Goldstern (TU Wien), Menachem   Kojman (Ben Gurion University, Israel)