Szemerédi theorem in a bounded-rank coset progression
= Szemerédi theorem in a bounded-rank coset progression
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For fixed density $\delta>0$, rank $r$, and progression length, every sufficiently large proper coset progression of rank at most $r$ has the property that each subset of relative density at least $\delta$ contains a nontrivial arithmetic progression of that length. This follows from the multidimensional <Szemerédi theorem>.