Friday, February 11, 2011 - 15:05
1 hour (actually 50 minutes)
We consider the question of coloring the first n integers with two colors in such a way as to avoid copies of a given arithmetic configuration (such as 3 term arithmetic progressions, or solutions to x+y=z+w). We know from results of Van der Waerden and others that avoiding such configurations completely is a hopeless task if n is sufficiently large, so instead we look at the question of finding colorings with comparatively few monochromatic copies of the configuration. At the very least, can we do significantly better than just closing our eyes and coloring randomly? I will discuss some partial answers, experimental results, and conjectured answers to these questions for certain configurations based on joint work with Steven Butler and Ron Graham.