Subsums of a Zero-sum Free Subset of An Abelian Group

报告题目:Subsums of a Zero-sum Free Subset of An Abelian Group

报 告 人:高维东 教授(南开大学)




Let G be an additive finite abelian group and S be a subset of G. Let sigma(S) be the sum of elements in S and Sigma(S) be the set of sigma(T) for T run through all nonempty subsets of S.We say that S is zero-sum free if 0 of G is not contained in Sigma(S). For posotive integer k, let f(G,k)min{Sigma(S): S is zero-sum free subsets of G with k elements}.

If all subsets of G with k elements are not zero-sum free, then set f(G,k)infinity. Let f(k)min{f(G,k):G is finite abelian group}.The invariant f(k) was first studied by R.B.Eggleton and P.Erdos in 1972. They determined the exact values of f(k) for k<5 and showed that 2k<f(k)<1 integer part of k^2/2 for k>4. In 1975, J.E. Olsen proved that f(k)>k^2/9, which was still the best known result on the lower bound of f(k) for large k(>27).

In this talk, we will give a short survey on the known values of f(k) and also present some recent developments in this area, including a much better lower bound for f(k).