  Let $r_k(n)$ denote the maximum cardinality of a set $A \subset \{1,2, \dots,
n \}$ such that $A$ does not contain a $k$-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.
