Monochromatic and zero-sum sets of nondecreasing diameter Original Research Article

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 B1, B2 ⊆ 1, 2, …, n (but not necessarily of the same color) which satisfy: (i) |B1¦=|B2|=m; (ii) The largest number in B1 is smaller than the smallest number in B2; (iii) The diameter of the convex hull spanned by B1 does not exceed the diameter of the convex hull spanned by B2. We prove that f(m, 2)=5m-3,f(m, 3)=9m-7 and 12m-9⩽f (m, 4) ⩽13m-11. Asymptotically, it is shown that e1mr⩽f(m,r)⩽c2mr log2 r, where c1 and c2 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 Erdős—Ginzburg—Ziv theorem. Moreover, stronger versions are derived when the group under consideration is cyclic of prime order.