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On sets of integers containing no \(k\) elements in arithmetic progression. (English) Zbl 0303.10056
In this paper, the author, by an exceedingly ingenious method, settles in the affirmative the following famous conjecture:
For all \(\varepsilon\) and \(k\), there exists \(N(k,\varepsilon)\) such that if \(n>N(k,\varepsilon)\) and \(R\) is a set of natural numbers \(\subseteq [0,n)\) with \(| R|>\varepsilon n\), then \(R\) contains an arithmetic progression with \(k\) terms.
Previously the conjecture was settled for the case \(k=3\) by K. F. Roth in 1952 [J. Lond. Math. Soc. 28, 104–109 (1953; Zbl 0050.04002)] and for the case \(k=4\) by E. Szemerédi himself in 1967 [Acta Math. Acad. Sci. Hung. 20, 89–104 (1969; Zbl 0175.04301)]. The method of proof in the present paper, though very complicated, is ‘elementary’.

11B25 Arithmetic progressions
05D05 Extremal set theory
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