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

For positive integers m and r, one can easily show there exist integers N such that for every map ∆ : {1, 2, · · · , N} → {1, 2, · · · , r} there exist 2m integers x1 < · · · < xm < y1 < · · · < ym which satisfy: 1. ∆(x1) = · · · = ∆(xm), 2. ∆(y1) = · · · = ∆(ym), 3. 2(xm − x1) ≤ ym − x1. 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.