# The minimum number of monochromatic 4-term progressions

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Julia Wolf, Rutgers University
Fine Hall 224

It is not difficult to see that whenever you 2-color the elements of $Z/pZ$, the number of monochromatic 3-term arithmetic progressions depends only on the density of the color classes. The analogous statement for 4-term progressions is false. We shall analyse the reasons for this, and subsequently derive bounds on the minimum number of monochromatic 4-term arithmetic progressions in any 2-coloring 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 2-coloring of $K_n$?