Abstract
When every set in a setpartition 
has size one, the DeVos-Goddyn-Mohar (DGM) Theorem gives a lower bound for |Σ
n
(S)|, where 
is the sequence partitioned by the setpartition 
. As seen in previous exercises, this is really a lower bound for 
, where the union runs over all n-setpartitions 
with 
. In other words, the DGM theorem, in this special case, gives a lower bound for the size of the union of all sumsets of the n-setpartitions of S. In this chapter, we present two versions of a sibling result to the DGM Theorem. Both essentially give the existence of a single
n-setpartition 
with large sumset. The first is stronger in statement and generalizes the aforementioned case of the DGM Theorem. The second is weaker in statement, but also valid in the much more general context of homomorphism weighted subsequence sums.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
A. Bialostocki, P. Dierker, D.J. Grynkiewicz, M. Lotspeich, On some developments of the Erdős-Ginzburg-Ziv theorem II. Acta Arith. 110(2), 173–184 (2003)
Y. Caro, Zero-sum problems—a survey. Discrete Math. 152(1–3), 93–113 (1996)
D.J. Grynkiewicz, On four colored sets with nondecreasing diameter and the Erdős-Ginzburg-Ziv Theorem. J. Comb. Theory, Ser. A 100(1), 44–60 (2002)
D.J. Grynkiewicz, An extension of the Erdős-Ginzburg-Ziv theorem to hypergraphs. Eur. J. Comb. 26(8), 1154–1176 (2005)
D.J. Grynkiewicz, On a conjecture of Hamidoune for subsequence sums. Integers 5(2), A7 (2005) (electronic)
D.J. Grynkiewicz, Sumsets, Zero-Sums and Extremal Combinatorics. Dissertation, Caltech, 2005
D.J. Grynkiewicz, On a partition analog of the Cauchy-Davenport theorem. Acta Math. Hung. 107(1–2), 161–174 (2005)
D.J. Grynkiewicz, A weighted Erdős-Ginzburg-Ziv Theorem. Combinatorica 26(4), 445–453 (2006)
D.J. Grynkiewicz, R. Sabar, Monochromatic and zero-sum sets of nondecreasing modified diameter. Electron. J. Comb. 13(1), Research Paper 28 (2006) (electronic)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Grynkiewicz, D.J. (2013). The Partition Theorem I. In: Structural Additive Theory. Developments in Mathematics, vol 30. Springer, Heidelberg. https://doi.org/10.1007/978-3-319-00416-7_14
Download citation
DOI: https://doi.org/10.1007/978-3-319-00416-7_14
Publisher Name: Springer, Heidelberg
Print ISBN: 978-3-319-00415-0
Online ISBN: 978-3-319-00416-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)