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

Haiyan Zhang

Department of Mathematics, Harbin University of Science and Technology, Harbin 150080, P. R. China

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and

Guoqing Wang

Department of Mathematics, Tianjin Polytechnic University, Tianjin 300387, P. R. China

Corresponding author.

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https://doi.org/10.1142/S1793042114500481Cited by:1 (Source: Crossref)

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 be the collection of all r-uniform intersecting families of size m. Let be the smallest positive integer d such that any G-coloring of the edges of the complete r-uniform hypergraph yields a zero-sum copy of some intersecting family in . Among other results, we mainly prove that , where Ω(s_m(G)) denotes the least positive integer n such that , and we show that if r | Ω(s_m(G)) – 1 then .

Keywords:

AMSC: 11B75, 05D10, 05C65

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