The size of subset of {1,…,N} without 3-term arithmetic progressions is O(N/(log n)^(1-o(1)))

You need to know: A (nontrivial) 3-term arithmetic progression, big O notation, small o notation.

Background: For integer $N>0$ , let $r_3(N)$ be the cardinality of the largest subset of $\{1, 2, \dots , N\}$ which contains no nontrivial 3-term arithmetic progressions.

The Theorem: On 30th October 2010, Tom Sanders submitted to arxiv a paper in which he proved that $r_3(N) = O\left(\frac{N(\log\log N)^6}{\log N}\right)$ .

Short context: In 1936, Erdős and Turán conjectured that any set containing a positive proportion of integers must contain a 3-term arithmetic progression. This is equivalent to $\lim\limits_{N\to\infty}\frac{r_3(N)}{N}=0$ . In 1953, Roth confirmed this conjecture by proving that $r_3(N) = O\left(\frac{N}{\log\log N}\right)$ . In 1999, Bourgain improved this to $r_3(N) = O\left(\frac{N\sqrt{\log\log N}}{\sqrt{\log N}}\right)$ . The Theorem proves a much stronger upper bound, the first one in the form $r_3(N) = O\left(\frac{N}{\log^{1-o(1)} N}\right)$ . In a later work, Bloom improved the bound slightly to $r_3(N) = O\left(\frac{N(\log\log N)^4}{\log N}\right)$ .