Erdős's similarity conjecture

Let $X\subseteq\mathbb{R}$ be a measure universal set, meaning that every Lebesgue measurable subset of $\mathbb{R}$ with positive Lebesgue measure contains an affine copy of $X$. Erdős's similarity conjecture. No infinite subset of the real line is measure universal. This is the original Erdős similarity problem; finite sets are measure universal, while the conjecture remains open for arbitrary uncountable sets and even for perfect sets.