Speaker:  Frederick Manners  (UCSD)

Title:  True Complexity and iterated Cauchy-Schwarz arguments

Abstract:

Lots of good things in combinatorics can be achieved by applying the Cauchy--Schwarz inequality several times.  For instance, we know that if a graph has density $\alpha$ and $(1+o(1))\alpha^4 n^4$ 4-cycles then it has $(1+o(1)) \alpha^{15} n^{10}$ copies of the Petersen graph.

On the arithmetic side, we know that if a set $A \subseteq G$ has density $\alpha$ and no large non-trivial Fourier coefficients, it contains -- for instance -- $(1+o(1)) \alpha^3 |G|^2$ three-term progressions $x,x+a,x+2a$.
But does it also contain $(1+o(1)) \alpha^6 |G|^3$ configurations

$x,x+z,x+y,x+y+z,x+2y+3z,2x+3y+6z$
or
$x,x+z,x+y,x+y+z,x+2y+3z,13x+12y+9z?$

Moreover, can you prove this by applying Cauchy--Schwarz multiple times?  If so, how do you find such an argument, and how horrifyingly long and complicated does it need to be?

We will discuss some answers to these questions as well as some open questions.