Large subsets of [Formula: see text] 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 [Formula: see text] . Let [Formula: see text] denote the maximal size of a subset of [Formula: see text] without arithmetic progressions of length k and let [Formula: see text] denote the least prime factor of m. We construct explicit progression-free sets and obtain the following improved lower bounds for [Formula: see text] : If [Formula: see text] is odd and [Formula: see text] , then [Formula: see text] If [Formula: see text] is even, [Formula: see text] and [Formula: see text] , then [Formula: see text] Moreover, we give some further improved lower bounds on [Formula: see text] for primes [Formula: see text] and progression lengths [Formula: see text] .