Improve the upper bound when face-sharing is disallowed

Determine whether forbidding face-sharing between rectangles in Mondrian partitions of an n × n square yields a better asymptotic upper bound on the minimum defect.

Background

The paper observes that the largest shared face between adjacent rectangles, or between rectangles sharing a dimension, often limits the size of tiles in computational searches for optimal Mondrian partitions. Face-sharing restricts the available distinct factorizations, and a defect-zero partition could not contain face-sharing because two equal-area tiles sharing a face would have the same dimension, which is prohibited. The authors therefore ask whether eliminating face-sharing can improve the paper's O(n{5/6}) upper bound on the defect.

References

Can you get a better upper bound if face-sharing is disallowed?

— New Upper bounds on the Mondrian Art Problem  (2609.01998 - Garrison et al., 2 Sep 2026) in Section 'Concluding remarks'

Another open question we have is: Are there more optimal algorithms that give a better upper bound on the defect?

— New Upper bounds on the Mondrian Art Problem  (2609.01998 - Garrison et al., 2 Sep 2026) in Section 'Concluding remarks'