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

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 Erdős-Ginzburg-Ziv Theorem, which is the case when H is a single edge.