Improved Extractors for Affine Lines [article]

Ariel Gabizon
2013 arXiv   pre-print
Let F be the field of q elements. We investigate the following Ramsey coloring problem for vector spaces: Given a vector space ^n, give a coloring of the points of F^n with two colors such that no affine line (i.e., affine subspace of dimension 1) is monochromatic. Our main result is as follows: For any q≥ 25· n and n>4, we give an explicit coloring D:F^n0,1 such that for every affine line l⊆ F^n, D(l)=0,1. Previously this was known only for q≥ c· n^2 for some constant c GR05. We note that this
more » ... beats the random coloring for which the expected number of monochromatic lines will be 0 only when q≥ c· n n for some constant c. Furthermore, our coloring will be 'almost balanced' on every affine line. Let us state this formally in the lanuage of extractors. We say that a function D:F^n0,1 is a 1 if for every affine line l⊆^n, D(X) is -close to uniform when X is uniformly distributed over l. We construct a 1 with = Ω(√(n/q)) whenever q≥ c· n for some constant c. The previous result of GR05 gave a 1 only for q=Ω(n^2).
arXiv:1311.5622v2 fatcat:i24dtl7mrbbjjjw5ndzuvujfxe