On a modification of a problem of Bialostocki, Erdős, and Lefmann

In this paper we investigate the minimal such integer, which we call g(m,r). We prove that g(m,2)=5(m-1)+1 for m⩾2, that g(m,3)=7(m-1)+1+⌈m/2⌉ for m⩾4, and that g(m,4)=10(m-1)+1 for m⩾3. Furthermore, we consider g(m,r) for general r. Along with results that bound g(m,r), we compute g(m,r) exactly for the following infinite families of r:{f2n+3},{2f2n+3},{18f2n-7f2n-2}and{23f2n-9f2n-2},where here fi is the ith Fibonacci number defined by f0=0 and f1=1.