5 research outputs found
Effectiveness of Hindman's theorem for bounded sums
We consider the strength and effective content of restricted versions of
Hindman's Theorem in which the number of colors is specified and the length of
the sums has a specified finite bound. Let denote the
assertion that for each -coloring of there is an infinite
set such that all sums for and have the same color. We prove that there is a
computable -coloring of such that there is no infinite
computable set such that all nonempty sums of at most elements of
have the same color. It follows that is not provable
in and in fact we show that it implies in
. We also show that there is a computable instance of
with all solutions computing . The proof of this
result shows that implies in