The de Bruijn–Erdős theorem states that the chromatic number of an infinite graph
equals the maximum of the chromatic numbers of its finite subgraphs. Such
determination by finite subobjects appears in the definition of a phantom map, which
is classical in algebraic topology. The topological method in combinatorics connects
these two, which leads us to define the relative version of a phantom map: a map
is called a relative
phantom map to a map
if the restriction of
to
any finite subcomplex of
lifts to
through
, up to homotopy.
There are two kinds of maps which are obviously relative phantom maps: (1) the composite of
a map
with
; (2) a usual
phantom map
.
A relative phantom map of type (1) is called trivial, and a relative phantom map out
of a suspension which is a sum of (1) and (2) is called relatively trivial. We study the
(relative) triviality of relative phantom maps and, in particular, we give rational
homology conditions for the (relative) triviality.
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