Canonization theorems for finite affine and linear spaces (Q760446)

In this paper, the author proves the following Theorem: let F be any finite field and let m be a positive integer. Then there exists a positive integer n such that, for every coloring \(\Delta: F^ n\to \omega\) of the affine points in \(F^ n\) with infinitely many colors, there exists an m-dimensional affine subspace A of \(F^ n\) such that the restriction of \(\Delta\) to A is either a constant coloring or a one to one coloring. He also proves several generalizations of this result.