  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.
