Monochromatic Progressions in Random Colorings
Combinatorics
2012-06-07 v2 Discrete Mathematics
Abstract
Let N^{+}(k)= 2^{k/2} k^{3/2} f(k) and N^{-}(k)= 2^{k/2} k^{1/2} g(k) where 1=o(f(k)) and g(k)=o(1). We show that the probability of a random 2-coloring of {1,2,...,N^{+}(k)} containing a monochromatic k-term arithmetic progression approaches 1, and the probability of a random 2-coloring of {1,2,...,N^{-}(k)} containing a monochromatic k-term arithmetic progression approaches 0, for large k. This improves an upper bound due to Brown, who had established an analogous result for N^{+}(k)= 2^k log k f(k).
Cite
@article{arxiv.1106.0793,
title = {Monochromatic Progressions in Random Colorings},
author = {Sujith Vijay},
journal= {arXiv preprint arXiv:1106.0793},
year = {2012}
}
Comments
5 pages