Bounded-- yes, but 4?

Abstract

In this paper we will examine unions of oriented and non-oriented unit squares in same plane and measure the ratio of perimeter to area of these unions. In 1998, T. Keleti published the conjecture that this ratio never exceeds 4. We outline the current state of research on this conjecture and give two proofs of a special case. Finally, we explore the difficulties that arise from using similar methods in the general case and examine properties of any potential counterexample.