On the Erdős-Ginzburg-Ziv invariant and zero-sum Ramsey number for intersecting families

2013/04/30 by Haiyan Zhang, Guoqing Wang, Zhang, Haiyan +1
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #math.CO #math.NT

paper · pdf · doi:10.48550/arxiv.1304.7957

14 papes

arxiv created 2013/04/30 · arxiv updated 2013/05/01

Abstract

Let G be a finite abelian group, and let m>0 with exp(G)| m. Let s_m(G) be the generalized Erdős-Ginzburg-Ziv invariant which denotes the smallest positive integer d such that any sequence of elements in G of length d contains a subsequence of length m with sum zero in G. For any integer r>0, let I_m^(r) be the collection of all r-uniform intersecting families of size m. Let R(I_m^(r),G) be the smallest positive integer d such that any G-coloring of the edges of the complete r-uniform hypergraph K_d^(r) yields a zero-sum copy of some intersecting family in I_m^(r). Among other results, we mainly prove that Ω(s_m(G))-1≤ R (I_m^(r), G)≤ Ω(s_m(G)), where Ω(s_m(G)) denotes the least positive integer n such that n-1 \choose r-1≥ s_m(G), and we show that if r| Ω(s_m(G))-1 then R (I_m^(r), G)= Ω(s_m(G)).

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