  Let $A\subset\left\{ 1,\dots,N\right\} $ be a set of prime numbers containing
no non-trivial arithmetic progressions. Suppose that $A$ has relative density
$\alpha=|A|/\pi(N)$, where $\pi(N)$ denotes the number of primes in the set
$\left\{ 1,\dots,N\right\} $. By modifying Helfgott and De Roton's work, we
improve their bound and show that $$\alpha\ll\frac{\left(\log\log\log
N\right)^{6}}{\log\log N}.$$
