Exploration of counter examples of balanced sets

Mathematicians are often intrigued with patterns, many times finding themselves looking for pieces of structure within a data set. This research project is no different in that we have explored our vast data set for substructure.Our goal is to identify the following: how many data points are necessary to guarantee our set has a balanced set/substructure? Naturally rephrasing the previous question, we also ask ourselves what is the largest set that does not have a balanced subset/substructure? This alternate phrasing set us down our current path. We focused on the largest sets with no balanced substructure and what they look like.After brute force checking all Z5 × Z5 maximal sets, we found 7 nonisomorphic graphs that did not have balanced substructure. Using those examples as starting points, we then extended into Z7 × Z7. When successful, our goal was to classify examples in Zp × Zp which have no balanced substructure.Currently, we believe there are four classifications of maximal sets with no balance substructure for any Zp × Zp. The main proof to follow focuses on one of these classifications called Kick It