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OPEN - $100

Let $A\subseteq\mathbb{R}$ be an infinite set. Must there be a set $E\subset \mathbb{R}$ of positive measure which does not contain any set of the shape $aA+b$ for some $a,b\in\mathbb{R}$ and $a\neq 0$?

The Erdős similarity problem.

This is true if $A$ is unbounded or dense in some interval. It therefore suffices to prove this when $A=\{a_1>a_2>\cdots\}$ is a countable strictly monotone sequence which converges to $0$.

Steinhaus [St20] has proved this is false whenever $A$ is a finite set.

This conjecture is known in many special cases (but, for example, it is is open when $A=\{1,1/2,1/4,\ldots\}$. For an overview of progress we recommend a nice survey by Svetic [Sv00] on this problem.

OPEN

Is there some $c>0$ such that every measurable $A\subseteq \mathbb{R}^2$ of measure $\geq c$ contains the vertices of a triangle of area 1?

Erdős (unpublished) proved that this is true if $A$ has infinite measure, or if $A$ is an unbounded set of positive measure.

OPEN

Let $A\subseteq \mathbb{R}^2$ be a measurable set with infinite measure. Must $A$ contain the vertices of an isosceles trapezoid of area $1$?

Erdős and Mauldin (unpublished) claim that this is true for trapezoids in general, but fails for parallelograms.