|Date: Friday, February 03, 2017
Location: 4088 East Hall (3:10 PM to 4:00 PM)
Title: Large colored sum free sets in finite vector spaces
Abstract: Let A be an abelian group. A colored sum free set of A is a list (a_1,b_1,c_1), (a_2,b_2,c_2), ..., (a_N,b_N,c_N) of triples of elements of A such that a_i+b_j+c_k=0 if and only if i=j=k. Extremal combinatorialists aim to construct large colored sum-free sets, both because it is fun and because it has applications in the construction of fast matrix multiplication algorithms. Note that, if X is a subset of A with no three term arithmetic progressions, then the set of (x,-2x,x) for x in X is a colored sum-free set, so bounds on colored sum-free sets are in particular bounds on sets without 3-term arithmetic progressions. Until May of 2016, the best such bounds were of the form A^(1-o(1)). Last May, Ellenberg and Gijswijt, building on work of Croot, Lev and Pach, proved bounds of the form A^c for c<1 when A is of the form (Z/p)^k. Robert Kleinberg, Will Sawin and I have constructed colored sum-free sets which meet these bounds to leading order. I will explain these exciting new bounds and our construction.
Speaker: David Speyer
Institution: U. Michigan