Avoiding Zero-sum Subsequences of Prescribed Length over the Integers
نویسندگان
چکیده
Let t and k be a positive integers, and let Ik = {i ∈ Z : −k ≤ i ≤ k}. Let st(Ik) be the smallest positive integer ` such that every zero-sum sequence S over Ik of length |S| ≥ ` contains a zero-sum subsequence of length t. If no such ` exists, then let st(Ik) =∞. In this paper, we prove that st(Ik) is finite if and only if every integer in [1, D(Ik)] divides t, where D(Ik) = max{2, 2k − 1} is the Davenport constant of Ik. Moreover, we prove that if st(Ik) is finite, then t + k(k − 1) ≤ st(Ik) ≤ t + (2k − 2)(2k − 3). We also show that st(Ik) = t + k(k − 1) holds for k ≤ 3 and conjecture that this equality holds for any k ≥ 1.
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Distinct Lengths Modular Zero-sum Subsequences: A Proof of Graham’s Conjecture
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