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On a combinatorial problem of Erdos, Kleitman and Lemke
In this paper, we study a combinatorial problem originating in the following
conjecture of Erdos and Lemke: given any sequence of n divisors of n,
repetitions being allowed, there exists a subsequence the elements of which are
summing to n. This conjecture was proved by Kleitman and Lemke, who then
extended the original question to a problem on a zero-sum invariant in the
framework of finite Abelian groups. Building among others on earlier works by
Alon and Dubiner and by the author, our main theorem gives a new upper bound
for this invariant in the general case, and provides its right order of
magnitude.Comment: 15 page
Zero-sum problems with congruence conditions
For a finite abelian group and a positive integer , let denote the smallest integer such that
every sequence over of length has a nonempty zero-sum
subsequence of length . We determine for all when has rank at most two and, under mild
conditions on , also obtain precise values in the case of -groups. In the
same spirit, we obtain new upper bounds for the Erd{\H o}s--Ginzburg--Ziv
constant provided that, for the -subgroups of , the Davenport
constant is bounded above by . This
generalizes former results for groups of rank two
The Large Davenport Constant I: Groups with a Cyclic, Index 2 Subgroup
Let be a finite group written multiplicatively. By a sequence over ,
we mean a finite sequence of terms from which is unordered, repetition of
terms allowed, and we say that it is a product-one sequence if its terms can be
ordered so that their product is the identity element of . The small
Davenport constant is the maximal integer such that
there is a sequence over of length which has no nontrivial,
product-one subsequence. The large Davenport constant is the
maximal length of a minimal product-one sequence---this is a product-one
sequence which cannot be factored into two nontrivial, product-one
subsequences. It is easily observed that , and
if is abelian, then equality holds. However, for non-abelian groups, these
constants can differ significantly. Now suppose has a cyclic, index 2
subgroup. Then an old result of Olson and White (dating back to 1977) implies
that if is non-cyclic, and
if is cyclic. In this paper, we determine the large Davenport constant of
such groups, showing that , where is the commutator subgroup of
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