On Sums of Distinct Representatives
نویسندگان
چکیده
Clearly (1) has an SDR provided that |Ai| > i for all i = 1, · · · , n, in particular an SDR of (1) exists if |A1| = · · · = |An| > n or 0 < |A1| < · · · < |An|. Let G be an additive abelian group and A1, · · · , An its subsets. We associate any SDR (2) of (1) with the sum ∑n i=1 ai and set (4) S({Ai}i=1) = S(A1, · · · , An) = {a1 + · · ·+ an : {ai}i=1 forms an SDR of {Ai}i=1} . Of course, S(A1, · · · , An) 6= ∅ if and only if (3) holds. A fascinating and challenging problem is to give a sharp lower bound for |S({Ai}i=1)| and determine when the bound can be reached. Let p be a prime. In 1964 P. Erdös and H. Heilbronn (cf. [EH] and [G]) conjectured that for each nonempty subset A of Zp = Z/pZ there are at least
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