The minimum number of monochromatic 4term progressions
The minimum number of monochromatic 4term progressions

Julia Wolf, Rutgers University
Fine Hall 224
It is not difficult to see that whenever you 2color the elements of $Z/pZ$, the number of monochromatic 3term arithmetic progressions depends only on the density of the color classes. The analogous statement for 4term progressions is false. We shall analyse the reasons for this, and subsequently derive bounds on the minimum number of monochromatic 4term arithmetic progressions in any 2coloring of $Z/pZ$. In the process we touch upon the subject of quadratic Fourier analysis as well as a closely related question in graph theory studied by Thomason et al.: What is the minimum number of monochromatic $K_4$s in any 2coloring of $K_n$?