Zero-sum over $\mathbb Z$

Zero-sum Ramsey theory is a newly established area in combinatorial number
theory. Zero-sum problems can be formulated as follows: If the elements of
a combinatorial structure are mapped into a finite group  $\Gamma$, does
there exists a prescribed substructure, such that the sum of the weights of
its elements is the neutral element of $\Gamma$?
Extending classical results in zero-sum Ramsey theory, we study mappings to
a set of integers $\{-r, \cdots ,0, \cdots ,r\}$  seeking zero-sum subsets.
We will focus in two types of structures, namely complete graphs and intervals
of integers.
Relying heavily on Pell equations and  some classical  biquadratic
Diophantine equations, we solve an open problem formulated by Caro and
Yuster and supply a good  understanding of the situation concerning the
complete graph $K_4$.
This is a joint work with Yair Caro and Adriana Hansberg.