On some developments of the Erdős–Ginzburg–Ziv Theorem II

Let S be a sequence of elements from the cyclic group Zm. We say S is zsf (zero-sum free) if there does not exist an m-term subsequence of S whose sum is zero. Denote by g(m, k) the least integer such that every sequence S with at least k distinct elements and length g(m, k) must contain an m-term subsequence whose sum is zero. Furthermore, denote by E(m, s) the set of all equivalence classes of zsf sequences with length s, up to order and affine transformation, that are not a proper subsequence of another zsf sequence. In this paper, we first find for a sequence S of sufficient length, |S| ≥ 2m − b 4 c − 2, necessary and sufficient conditions in terms of a system of inequalities over the integers for S to be zsf. Among the consequences, we determine g(m, k) for large m, namely g(m, k) = 2m− b k 2−2k+5 4 c provided m ≥ k − 2k − 3, which in turn resolves two conjectures of the first and fourth authors. Next, using independent methods, we evaluate g(m, 5) for every m ≥ 5. We conclude with the list of E(m, s) for every m and s satisfying 2m− 2 ≥ s ≥ max{2m− 8, 2m− b 4 c − 2}.

[1]  Oscar Ordaz,et al.  On the Erdös-Ginzburg-Ziv theorem , 1996, Discret. Math..

[2]  David J. Grynkiewicz,et al.  On a partition analog of the Cauchy-Davenport theorem , 2005 .

[3]  Yair Caro.  Remarks on a Zero-Sum Theorem , 1996 .

[4]  Weidong Gao,et al.  Zero Sums in Abelian Groups , 1998, Comb. Probab. Comput..

[5]  Oscar Ordaz,et al.  On a Combinatorial Theorem of Erdös, Ginzburg and Ziv , 1998, Comb. Probab. Comput..

[6]  ON SUMS OF DISTINCT REPRESENTATIVES , 1998 .

[7]  Yahya Ould Hamidoune,et al.  On Restricted Sums , 2000, Comb. Probab. Comput..

[8]  Peter M. Neumann.  Two combinatorial problems in group theory , 1989 .

[9]  Noga Alon,et al.  The Polynomial Method and Restricted Sums of Congruence Classes , 1996 .

[10]  Vsevolod F. Lev,et al.  Restricted set addition in groups , 2000 .

[11]  Weidong Gao.  Addition Theorems for Finite Abelian Groups , 1995 .

[12]  P. Erdos,et al.  Old and new problems and results in combinatorial number theory , 1980 .

[13]  Roger Crocker,et al.  A theorem in additive number theory , 1969 .

[14]  Harold Davenport,et al.  On the Addition of Residue Classes , 1935 .

[15]  Arie Bialostocki,et al.  On the Erdös-Ginzburg-Ziv theorem and the Ramsey numbers for stars and matchings , 1992, Discret. Math..

[16]  Vsevolod F. Lev.  Restricted Set Addition in Groups I: The Classical Setting , 2000 .

[17]  A. Ziv,et al.  Theorem in the Additive Number Theory , 2022 .

[18]  Melvyn B. Nathanson,et al.  Additive Number Theory , 1996 .

[19]  Luis H. Gallardo,et al.  On a variant of the Erdős-Ginzburg-Ziv problem , 1999 .

[20]  F. Hennecart.  Restricted Addition and some Developments of the Erdős–Ginzburg–ziv Theorem , 2005 .

[21]  Yahya Ould Hamidoune,et al.  A note on the minimal polynomial of the Kronecker sum of two linear operators , 1990 .

[22]  Werner Brakemeier.  Eine Anzahlformel von Zahlen modulon , 1978 .

[23]  Noga Alon,et al.  Zero-sum sets of prescribed size , 1993 .

[24]  J. H. B. Kemperman,et al.  On small sumsets in an abelian group , 1960 .

[25]  Alain Plagne,et al.  Restricted addition in Z/nZ and an Application to the Erdős–Ginzburg–Ziv Problem , 2002 .

[26]  Weidong Gao.  An addition theorem for finite cyclic groups , 1997, Discret. Math..