Erdős, P. and Turán, P. "On Some Sequences of Integers." J. London Math. Soc.11, 261-264, 1936.Fürstenberg, H. "Ergodic Behavior of Diagonal Measures and a Theorem of Szemerédi on Arithmetic Progressions." J. Analyse Math.31, 204-256, 1977.Fürstenberg, H. and Katznelson, Y. "An Ergodic Szemerédi Theorem for Commuting Transformations." J. Analyse Math.34, 275-291, 1979.Fürstenberg, H. and Weiss, B. "A Mean Ergodic Theorem for ." In Convergence in Ergodic Theory and Probability (Columbus OH 1993). Berlin: de Gruyter, pp. 193-227, 1996.Fürstenberg, H.; Katznelson, Y.; and Ornstein, D. "The Ergodic-Theoretical Proof of Szemerédi's Theorem." Bull. Amer. Math. Soc.7, 527-552, 1982.Gowers, W. T. "Fourier Analysis and Szemerédi's Theorem." In Proceedings of the International Congress of Mathematicians, Vol. I (Berlin, 1998).Doc. Math., Extra Vol. I, 617-629, 1998a.Gowers, W. T. "A New Proof of Szemerédi's Theorem for Arithmetic Progressions of Length Four." Geom. Funct. Anal.8, pp. 529-551, 1998b.Gowers, W. T. "A New Proof of Szemerédi's Theorem." Geom. Funct. Anal.11, 465-588, 2001.Graham, R. L.; Rothschild, B. L.; and Spencer, J. H. Ramsey Theory, 2nd ed. New York: Wiley, 1990.Green, B. and Tao, T. "The Primes Contain Arbitrarily Long Arithmetic Progressions." Preprint. 8 Apr 2004. http://arxiv.org/abs/math.NT/0404188.Guy, R. K. "Theorem of van der Waerden, Szemerédi's Theorem. Partitioning the Integers into Classes; at Least One Contains an A.P." §E10 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 204-209, 1994.Lepowsky, J.; Lindenstrauss, J.; Manin, Y.; and Milnor, J. "The Mathematical Work of the 1998 Fields Medalists." Not. Amer. Math. Soc.46, 17-26, 1999.Roth, K. "Sur quelques ensembles d'entiers." Comptes Rendus Acad. Sci. Paris234, 388-390, 1952.Roth, K. F. "On Certain Sets of Integers." J. London Math. Soc.28, 104-109, 1953.Szemerédi, E. "On Sets of Integers Containing No Four Elements in Arithmetic Progression." Acta Math. Acad. Sci. Hungar.20, 89-104, 1969.Szemerédi, E. "On Sets of Integers Containing No Elements in Arithmetic Progression." Acta Arith.27, 199-245, 1975a.Szemerédi, E. "On Sets of Integers Containing No Elements in Arithmetic Progression." In Proceedings of the International Congress of Mathematicians, Volume 2, Held in Vancouver, B.C., August 21-29, 1974. Montreal, Quebec: Canad. Math. Congress, pp. 503-505, 1975b.
和正实数 
，存在一个阈值数 
，使得对于 
，{1,2,...,n} 的每个 基数 大于 
的子集都包含一个 
-项 等差数列。范德瓦尔登定理 可以通过设置 
立即推导出。 范德瓦尔登数 的最佳界限是从塞迈雷迪定理中 
的界限推导出来的。
的情况，并在他的菲尔兹奖章引文中被提及。Szemerédi (1969) 证明了 
的情况，并在 1975 年证明了一般定理，这是 塞迈雷迪正则性引理 (Szemerédi 1975a) 的一个结果，为此他从 Erdos 那里获得了 1000 美元的奖金。Fürstenberg 和 Katznelson (1979) 使用 遍历理论 证明了塞迈雷迪定理。Gowers (1998ab) 随后给出了一个新的证明，对于 
的情况 
（在他的菲尔兹奖章引文中提到；Lepowsky et al. 1999），
." In 
Elements in Arithmetic Progression." Acta Arith. 27, 199-245, 1975a.Szemerédi, E. "On Sets of Integers Containing No 
Elements in Arithmetic Progression." In Proceedings of the International Congress of Mathematicians, Volume 2, Held in Vancouver, B.C., August 21-29, 1974. Montreal, Quebec: Canad. Math. Congress, pp. 503-505, 1975b.