Geometry Seminar

Hypergaph and Fractional Generalizations of the Erdős-Ginzburg-Ziv Theorem

Speaker: Steve Simon, Bard College

Location: Warren Weaver Hall 1314

Date: Tuesday, April 30, 2024, 6 p.m.

Synopsis:

A cornerstone result of Erdős, Ginzburg, and Ziv (EGZ) states that any sequence of \(2n-1\) elements in \(\mathbb{Z}/n\) contains a zero-sum subsequence of length \(n\). While algebraic techniques have predominated in deriving many deep generalizations of this theorem over the past sixty years, here we introduce topological approaches to zero-sum problems which have proven fruitful in other combinatorial contexts. Our main result  is a topological criterion for determining when any \(\mathbb{Z}/n\)-coloring ing of an -uniform hypergraph contains a zero-sum hyperedge. In addition to applications for Kneser hypergraphs, for complete hypergraphs our methods recover Olson's generalization of the EGZ theorem for arbitrary finite groups. 
In addition, we give a fractional generalization of the EGZ theorem with applications to balanced set families.

Notes:

In-person and on Zoom.  See mailing list announcements for Zoom details or contact Boris Aronov.