New Upper bounds on the Mondrian Art Problem

We present a new upper bound on the defect of the Mondrian Art Problem. The Mondrian Art Problem asks for a partition of an $n\times n$ square with rectangles of distinct dimensions such that the difference (defect) between the largest and smallest rectangle areas is minimized. We prove that for any $n\times n$ square, there exists a partition with defect $O(n^{5\mathrm{/}6})$, improving upon the previously conjectured $O(n\mathrm{/}\log n)$ upper bound. We also implement an algorithm that provides empirical evidence supporting our theoretical bound.