Szemeredi’s theorem holds with doubly exponential bound

You need to know: Integers, addition, multiplication, power, logarithm, set, subset, size of set.

Background: Here, by arithmetic progression of length k we mean a sequence $a_1, a_2, \dots, a_k$ such that $a_n = a_1 + (n-1)d$ , $n=1,2,\dots, k$ , for some $d \neq 0$ .

The Theorem: In March 2000, Timothy Gowers submitted to Geometric and Functional Analysis (GAFA) a paper in which he proved that, for every positive integer k, every subset of $\{1, 2, \dots ,N\}$ of size at least $N(\log \log N)^{-c}$ , where $c=2^{-2^{k+9}}$ , contains an arithmetic progression of length k.

Short context: In 1975, Szemerédi proved that for any positive integer k and any $\delta > 0$ , there exists a positive integer $N = N(k, \delta)$ such that every subset of the set $\{1, 2, \dots ,N\}$ of size at least $\delta N$ contains an arithmetic progression of length k. This famous theorem is one of the milestones of combinatorics. However, the bound for N which follows from Szemerédi’s proof is huge. The Theorem proves that Szemeredi’s theorem holds with $N=\exp(\exp(\delta^c))$ .