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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’.

##### MSC:
 11B25 Arithmetic progressions 05D05 Extremal set theory
##### Keywords:
arithmetic progressions
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