An extension of the Erdos-Ginzburg-Ziv Theorem to hypergraphs

An n-set partition of a sequence S is a collection of n nonempty subsequences of S, pairwise disjoint as sequences, such that every term of S belongs to exactly one of the subsequences, and the terms in each subsequence are all distinct with the result that they can be considered as sets. For a sequence S, subsequence S', and set T, |T ∩ S| denotes the number of terms x of S with x ∈ T, and |S| denotes the length of S, and S \ S' denotes the subsequence of S obtained by deleting all terms in S'. We first prove the following two additive number theory results.(1) Let S be a finite sequence of elements from an abelian group G. If S has an n-set partition, A = A1, ..., An, such that |Σi=1n Ai| ≥ Σi=1n |Ai| - n + 1, then there exists a subsequence S' of S, with length |S'| ≤ max{|S - n + 1, 2n}, and with an n-set partition, A' = A'1, ..., A'n, such that |Σi=1n A'i| ≥ Σi=1n |Ai| - n + 1. Furthermore, if ||Ai| - |Aj|| ≤ 1 for all i and j, or if |Ai| ≥ 3 for all i, then A'i ⊆ Ai.(2) Let S be a sequence of elements from a finite abelian group G of order m, and suppose there exist a, b ∈ G such that |(G \ {a, b}) ∩ S| ≤ ⌊m/2⌋. If |S| ≥ 2m - 1, then there exists an m-term zero-sum subsequence S' of S with |{a} ∩ S'| ≥ ⌊ m/2 ⌋ or |{b} ∩ S'| ≥ ⌊ m/2 ⌋.Let H be a connected, finite m-uniform hypergraph, and let f(H) (let fzs(H)) be the least integer n such that for every 2-coloring (coloring with the elements of the cyclic group Zm) of the vertices of the complete m-uniform hypergraph Knm, there exists a subhypergraph K isomorphic to H such that every edge in K is monochromatic (such that for every edge e in K the sum of the colors on e is zero). As a corollary to the above theorems, we show that if every subhypergraph H' of H contains an edge with at least half of its vertices monovalent in H', or if H consists of two intersecting edges, then fzs(H) = f(H). This extends the Erdos-Ginzburg-Ziv Theorem, which is the case when H is a single edge.

[1]  M. Kneser,et al.  Abschätzung der asymptotischen Dichte von Summenmengen , 1953 .

[2]  Xiang-dong Hou,et al.  A Generalization of an Addition Theorem of Kneser , 2002 .

[3]  Daniel Schaal,et al.  A zero-sum theorem , 2003, J. Comb. Theory, Ser. A.

[4]  M. Kneser,et al.  Ein Satz über abelsche Gruppen mit Anwendungen auf die Geometrie der Zahlen , 1954 .

[5]  Yair Caro,et al.  Zero-sum problems - A survey , 1996, Discret. Math..

[6]  David J. Grynkiewicz,et al.  Monochromatic and Zero-Sum Sets of Nondecreasing Modified Diameter , 2006, Electron. J. Comb..

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

[8]  David J. Grynkiewicz,et al.  Quasi-periodic decompositions and the Kemperman structure theorem , 2005, Eur. J. Comb..

[9]  E. Haacke.  Sequences , 2005 .

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

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

[12]  David J. Grynkiewicz,et al.  On Four Colored Sets with Nondecreasing Diameter and the Erds-Ginzburg-Ziv Theorem , 2002, J. Comb. Theory, Ser. A.

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

[14]  Melvyn B. Nathanson,et al.  Additive Number Theory: Inverse Problems and the Geometry of Sumsets , 1996 .

[15]  K. Brown,et al.  Graduate Texts in Mathematics , 1982 .

[16]  Paul Erdös,et al.  Monochromatic and zero-sum sets of nondecreasing diameter , 1995, Discret. Math..

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

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

[19]  Paul D. Seymour,et al.  A simpler proof and a generalization of the zero-trees theorem , 1991, J. Comb. Theory, Ser. A.

[20]  Zoltán Füredi,et al.  On zero-trees , 1992, J. Graph Theory.

[21]  S. S. Pillai.  On the addition of residue classes , 1938 .

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

[23]  David J. Grynkiewicz,et al.  On some developments of the Erdős–Ginzburg–Ziv Theorem II , 2003 .