An extension of the Erdős–Ginzburg–Ziv Theorem to hypergraphs

Abstract

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 = A 1 , … , A n , such that | ∑ i = 1 n A i | ≥ ∑ i = 1 n | A i | − n + 1 , then there exists a subsequence S ′ of S , with length | S ′ | ≤ max { | S | − n + 1 , 2 n } , and with an n -set partition, A ′ = A 1 ′ , … , A n ′ , such that | ∑ i = 1 n A i ′ | ≥ ∑ i = 1 n | A i | − n + 1 . Furthermore, if | | A i | − | A j | | ≤ 1 for all i and j , or if | A i | ≥ 3 for all i , then A i ′ ⊆ A i . (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 | ≥ 2 m − 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 f z s ( H ) ) be the least integer n such that for every 2-coloring (coloring with the elements of the cyclic group Z m ) of the vertices of the complete m -uniform hypergraph K n m , 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 f z s ( H ) = f ( H ) . This extends the Erdős–Ginzburg–Ziv Theorem, which is the case when H is a single edge.