Extensions and generalizations[edit]

A multidimensional generalization of Szemerédi's theorem was first proven by Hillel Furstenberg and Yitzhak Katznelson using ergodic theory.^[27] Timothy Gowers,^[28] Vojtěch Rödl and Jozef Skokan^[29][30] with Brendan Nagle, Rödl, and Mathias Schacht,^[31] and Terence Tao^[32] provided combinatorial proofs.


Alexander Leibman and Vitaly Bergelson^[33] generalized Szemerédi's to polynomial progressions: If ${\displaystyle A\subset \mathbb {N} }$ is a set with positive upper density and ${\displaystyle p_{1}(n),p_{2}(n),\dotsc ,p_{k}(n)}$ are integer-valued polynomials such that ${\displaystyle p_{i}(0)=0}$, then there are infinitely many ${\displaystyle u,n\in \mathbb {Z} }$ such that ${\displaystyle u+p_{i}(n)\in A}$ for all ${\displaystyle 1\leq i\leq k}$. Leibman and Bergelson's result also holds in a multidimensional setting.


The finitary version of Szemerédi's theorem can be generalized to finite additive groups including vector spaces over finite fields.^[34] The finite field analog can be used as a model for understanding the theorem in the natural numbers.^[35] The problem of obtaining bounds in the k=3 case of Szemerédi's theorem in the vector space ${\displaystyle \mathbb {F} _{3}^{n}}$ is known as the cap set problem.


The Green–Tao theorem asserts the prime numbers contain arbitrarily long arithmetic progressions. It is not implied by Szemerédi's theorem because the primes have density 0 in the natural numbers. As part of their proof, Ben Green and Tao introduced a "relative" Szemerédi theorem which applies to subsets of the integers (even those with 0 density) satisfying certain pseudorandomness conditions. A more general relative Szemerédi theorem has since been given by David Conlon, Jacob Fox, and Yufei Zhao.^[36][37]


The Erdős conjecture on arithmetic progressions would imply both Szemerédi's theorem and the Green–Tao theorem.