We prove that if A ⊆ { 1 , … N } does not contain any non-trivial three-term arithmetic progression, then | ≪ ( log ⁡ ) 3 + o .

Abstract We show that there is a red-blue colouring of $[N]$ with no blue 3-term arithmetic progression and red length $e^{C(\log N)^{3/4}(\log \log N)^{1/4}}$ . Consequently, the two-colour van der Waerden number $w(3,k)$ bounded below by $k^{b(k)}$ , where $b(k) = c \big ( \frac {\log k}{\log k} )^{1/3}$ Previously it had been speculated, supported data, $w(3,k) O(k^2)$

In this list there is an arithmetic progression: 1, 25, 49 (common difference 24). If we search further along, another arithmetic progression of squares is found: 289, 625, 961 (common difference 336). Yet another is 529, 1369, 2209 (common difference 840). How can these examples, and all others, be found? In Section 2 we will use plane geometry to describe the 3-term arithmetic progressions of...

Addressing a question of Cameron and Erdős, we show that, for infinitely many values of n, the number of subsets of {1, 2, . . . , n} that do not contain a k-term arithmetic progression is at most 2O(rk(n)), where rk(n) is the maximum cardinality of a subset of {1, 2, . . . , n} without a k-term arithmetic progression. This bound is optimal up to a constant factor in the exponent. For all value...

Consider natural numbers {1, · · · , n} colored in three colors. We prove that if each color appears on at least (n + 4)/6 numbers then there is a three-term arithmetic progression whose elements are colored in distinct colors. This variation on the theme of Van der Waerden’s theorem proves the conjecture of Jungić et al.

In this paper, we prove that there is an arithmetic progression of positive odd numbers for each term M of which none of five consecutive odd numbers M,M − 2,M − 4,M − 6 and M − 8 can be expressed in the form 2n ± pα, where p is a prime and n, α are nonnegative integers.