Monochromatic and zero-sum sets of nondecreasing diameter

For positive integers m and r define f (m, r) to be the minimum integer n such that for every coloring of {1,2 . . . . . n} with r colors, there exist two monochromatic subsets B 1, B2~{1, 2 . . . . . n} (but not necessarily of the same color) which satisfy: (i)IBll=lB21=m; (ii) The largest number in B 1 is smaller than the smallest number in B2; (iii) The diameter of the convex hull spanned by B 1 does not exceed the diameter of the convex hull spanned by B z. We prove that f(m, 2) = 5m 3,f(m, 3) = 9 m 7 and 12m 9 <~f(m, 4) ~< 13m1 l. Asymptotically, it is shown that cxmr<~f(m,r)<~c2mrlog2r, where c 1 and c 2 are positive constants. Next we consider the corresponding questions for zero-sum sets and we generalize some of our results in the sense of the Erdds-Ginzburg-Ziv theorem. Moreover, stronger versions are derived when the group under consideration is cyclic of prime order.