Small Sets which meet all the n -Term Arithmetic Progressions in the Interval [1, n2 ]
JOHN KENNETH TRUSS · Bulletin of the London Mathematical Society · 1991
For any positive integers n and k, let f(n, k) denote the smallest size of a subset of the integer interval I = [l, n] which meets all the k-term arithmetic progressions contained in I. We show that n + ( 1 / 2 ) n 1 / 2 − 2 < f ( n 2 , n ) ⩽ p + ⌈ ( n − 1 ) 2 p ⌉ , where p is the largest prime ⩽ n, and for any real number x, [x] is the least integer ⩾ x.