A Constructive Lower Bound on Szemer\'edi's Theorem

Let $r_k(n)$ denote the maximum cardinality of a set $A \subset {1,2, \dots, n }$suchthat$A$doesnotcontaina$k$-term arithmetic progression. In this paper, we give a method of constructing such a set and prove the lower bound $n^{1-\frac{c_k}{k\ln k}}<r_k(n)$ where $k$ is prime, and $c_k \rightarrow 1$as$k \rightarrow \infty$. This bound is the best known for an increasingly large interval of $n$ as we choose larger and larger $k$. We also demonstrate that one can prove or disprove a conjecture of Erd\H{o}s on arithmetic progressions in large sets once tight enough bounds on $r_k(n)$ are obtained.