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和正實數 
，存在一個閾值數 
，使得對於 
，{1,2,...,n} 的每個 基數 大於 
的子集都包含一個 
-項 等差數列。範德瓦爾登定理 可以透過設定 
立即推匯出。 範德瓦爾登數 的最佳界限是從塞邁雷迪定理中 
的界限推匯出來的。
的情況，並在他的菲爾茲獎章引文中被提及。Szemerédi (1969) 證明了 
的情況，並在 1975 年證明了一般定理，這是 塞邁雷迪正則性引理 (Szemerédi 1975a) 的一個結果，為此他從 Erdos 那裡獲得了 1000 美元的獎金。Fürstenberg 和 Katznelson (1979) 使用 遍歷理論 證明了塞邁雷迪定理。Gowers (1998ab) 隨後給出了一個新的證明，對於 
的情況 
（在他的菲爾茲獎章引文中提到；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.