Fodor-type Reflection Principle (FRP) is the assertion that the following
FRP(kappa) holds for all regular cardinals kappa>aleph_1:
FRP(kappa): For any stationary subset E of E^kappa_omega and
g:E -> [kappa]^aleph0, there is an I in [kappa]^aleph1 such that
(1) cf(I)=omega_1;
(2) I is closed with respect to g; and
(3) for any f:E intersect I -> kappa, if f(alpha) in g(alpha) intersect alpha
for all alpha in E intersect I, then there is a beta* in I such that
f^{-1}``{beta*} is stationary in sup(I).
Using a new characterization of FRP we show that many reflection theorems
originally obtained as consequences of Axiom R are actually
equivalent to FRP over ZFC. The following two are among such assertions
equivalent to FRP:
-- For every locally countably compact topological space X, if all
subspaces of X of cardinality leqaleph_1 are metrizable, then X
itself is metrizable.
-- For any graph G, if all subgraphs of G of cardinality leqaleph_1
have countable coloring number, then G itself has countable coloring
number.
The main results of this talk are obtained in a joint
research with Lajos Soukup, Hiroshi Sakai and Toshimichi Usuba.