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Abstract: George Szekeres described some subsets of without arithmetic progressions of length p for odd primes p, obtained by a greedy algorithm. Let denote the size of the largest subset of 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 and van der Waerden numbers are derived by generalizing these constructions. In particular, for any odd prime p, we prove that , where . [Copyright &y& Elsevier]

Abstract: For positive integers n and k, let be the size of the largest subset of without arithmetic progressions of length k. The van der Waerden number is the smallest integer w such that every r-coloring of contains a monochromatic -term arithmetic progression with color i for some i. In this note, an algorithm is proposed to search exact values of for some k and n, and some new exact values of for are obtained. The results extend the previous ones significantly. It is also shown that for prime , and three lower bounds for van der Waerden numbers are given: , , . [Copyright &y& Elsevier]