Ginzburg Ziv Theorem Research Articles

Remarks on a Zero-Sum Theorem

Recently the following theorem in combinatorial group theory has been proved: LetGbe a finite abelian group and letAbe a sequence of members ofGsuch that |A|⩾|G|+D(G)−1, whereD(G) is the Davenport constant ofG. ThenAcontains a subsequenceBsuch that |B|=|G| and ∑b∈Bb=0. We shall present a generalization of this theorem which contains information on the extremal cases and in particular allows us to deduce a short proof of the extremal cases in the Erdős–Ginzburg–Ziv theorem. We also present, using the above-mentioned theorem, a proof that ifGhas rankkthen |A|⩾|G|(1+(k+1)/2k)−1 suffices to ensure a zero-sum subsequence on |G| terms.

On Weighted Sequence Sums

The main result of this paper has the following consequence. Let G be an abelian group of order n. Let {xi: 1 ≤ 2n − 1} be a family of elements of G and let {wi: 1 ≤ i ≤ n − 1} be a family of integers prime relative to n. Then there is a permutation & of [1,2n − 1] such thatApplying this result with wi = 1 for all i, one obtains the Erdős–Ginzburg–Ziv Theorem.

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 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 log2r, 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.