Abstract
George Szekeres described some subsets of { 1 , … , n } without arithmetic progressions of length p for odd primes p , obtained by a greedy algorithm. Let r k ( n ) denote the size of the largest subset of { 1 , … , n } without arithmetic progressions of length k . In this paper, the history of results based on the constructions by Szekeres is briefly surveyed. New inequalities for r k ( n ) and van der Waerden numbers are derived by generalizing these constructions. In particular, for any odd prime p , we prove that r p ( p 2 ) ⩾ ( p − 1 ) 2 + t p , where lim p → ∞ t p ln p = 1 .
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