= Furstenberg-Katznelson theorem
{c}
{title2=$a+tF\subseteq A$}
The multidimensional <Szemerédi theorem> says that a subset $A\subseteq\mathbb Z^d$ with positive upper density contains a translated positive integer dilate $a+tF$ of every finite set $F\subseteq\mathbb Z^d$. In the finite formulation, for every $\alpha>0$ and finite $F$, all sufficiently large integer boxes have this property for every subset occupying at least an $\alpha$ fraction of the box. The one-dimensional case for $F=\{0,1,\ldots,r-1\}$ is the <Szemerédi theorem>. The general result follows from multiple recurrence for commuting measure-preserving transformations.
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