Monochromatic progressions in random colorings

Let N + ( k ) = 2 k / 2 k 3 / 2 f ( k ) and N − ( k ) = 2 k / 2 k 1 / 2 g ( k ) where f ( k ) → ∞ and g ( k ) → 0 arbitrarily slowly as k → ∞ . 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, as k → ∞ . This improves an upper bound due to Brown, who had established an analogous result for N + ( k ) = 2 k log k f ( k ) .