The covering problem of Rado

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $\mathcal C$ be a nonempty finite collection of closed axis-parallel squares of positive side length in $\mathbb R^2$, with side lengths allowed to differ. Write $|E|$ for planar Lebesgue area. A subcollection $\mathcal S\subseteq\mathcal C$ is disjoint if $Q\cap Q'=\varnothing$ whenever $Q,Q'\in\mathcal S$ are distinct; boundary contact therefore counts as intersection. Define \[M(\mathcal C)=\max_{\substack{\mathcal S\subseteq\mathcal C\\\mathcal S\text{ disjoint}}}\left|\bigcup_{Q\in\mathcal S}Q\right|,\qquad c_\square=\inf_{|\bigcup_{Q\in\mathcal C}Q|=1}M(\mathcal C).\]

Statement recorded in the supplied source.

Determine $c_\square$: how small can the largest area of a disjoint subcollection be when the union of the original squares has area one? Equivalently, determine the largest universal $c$ for which every finite collection satisfies $M(\mathcal C)\geq c\,|\bigcup_{Q\in\mathcal C}Q|$. [S2]

Short English statement

Among finite collections of axis-parallel squares covering area one, determine the smallest possible maximum area that can be retained by selecting pairwise disjoint squares.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 7200025 (AMR-071-0025). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Gian Maria Dall’Ara and Adrian Dumitrescu, Rado’s covering problem for cubes and balls: a semi-survey (2026). Section 2, equation (3), pp. 2–3. https://arxiv.org/pdf/2604.17509v2.