A note on the Erd\H{o}s conjecture about square packing

TL;DR

This paper proves Erdős's conjecture on the maximum total side length of squares packed inside a unit square for the case when the squares are aligned with the container, confirming the conjecture under this condition.

Contribution

The paper establishes the validity of Erdős's conjecture for aligned squares, providing a proof under the assumption of parallel sides.

Findings

Erdős's conjecture holds for aligned squares.

Maximum total side length is exactly k for k^2+1 squares.

The result confirms the conjecture in the aligned case.

Abstract

Let $f(n)$ denote the maximum total length of the sides of $n$ squares packed inside a unit square. Erd\H{o}s conjectured that $f(k^2+1)=k$. We show that the conjecture is true if we assume that the sides of the squares are parallel to the sides of the unit square.