Large subsets of  without arithmetic progressions

For integers m and n, we study the problem of finding good lower bounds for the size of progression-free sets in $$ . Let $$ denote the maximal size of a subset of $$ without arithmetic progressions of length k and let $$ denote the least prime factor of m. We construct explicit progression-free sets and obtain the following improved lower bounds for $$ :If $$ is odd and $$ , then If $$ is even, $$ and $$ , then Moreover, we give some further improved lower bounds on $$ for primes $$ and progression lengths $$ .