An infinitary Zero sum theorem

TL;DR

This paper extends the Erdős-Ginzburg-Ziv theorem to an infinitary setting using ultrafilter methods, connecting additive combinatorics with Ramsey theory.

Contribution

It introduces an infinitary version of the Erdős-Ginzburg-Ziv theorem and demonstrates its proof via ultrafilter techniques, linking it to Ramsey theory.

Findings

Established an infinitary zero-sum theorem.

Connected the theorem with Ramsey-theoretic large sets.

Used ultrafilter methods in the proof.

Abstract

Erd\H{o}s-Ginzburg-Ziv theorem says that if there are 2n-1 number is given, then there are n numbers such that their sum is divided by n. We will connect this theorem with the Ramsey theoretic large sets and will prove an infinitary version of this theorem. In our proof we will use the methods of ultrafilters. But one may proceed using methods of Topological dynamics.