On sets of integers containing k elements in arithmetic progression

In 1926 van der Waerden [13] proved the following startling theorem : If the set of integers is arbitrarily partitioned into two classes then at least one class contains arbitrarily long arithmetic progressions. It is well known and obvious that neither class must contain an infinite arithmetic progression. In fact, it is easy to see that for any sequence an there is another sequence bn9 with bn > an9 which contains no arithmetic progression of three terms, but which intersects every infinite arithmetic progression. The finite form of van der Waerden's theorem goes as follows: For each positive integer n9 there exists a least integer f{n) with the property that if the integers from 1 to /(/?) are arbitrarily partitioned into two classes, then at least one class contains an arithmetic progression of « terms. (For a short proof, see the note of Graham and Rothschild [5].) However, the best upper bound on f{n) known at present is extremely poor. The best lower bound known, due to Berlekamp [3], asserts that/(«) < nl9 for n prime, which improves previous results of Erdös, Rado and W. Schmidt. More than 40 years ago, Erdös and Turân [4] considered the quantity rk{n)9 defined to be the greatest integer / for which there is a sequence of integers 0 < a\ < a2 < ••• < a; ^ n which does not contain an arithmetic progression of k terms. They were led to the investigation of rk{n) by several things. First of all the problem of estimating rk{n) is clearly interesting in itself. Secondly, rk{n) < n/2 would imply f{k) < 77, i.e., they hoped to improve the poor upper bound on f{k) by investigating rk{n). Finally, an old question in number theory asks if there are arbitrarily long arithmetic progressions of prime numbers. From rk{n) < %{rì) this would follow immediately. The hope was that this problem on primes could be attacked not by