2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05)
Stefan Felsner (ed.)
DMTCS Conference Volume AE (2005), pp. 323328
author:  Benjamin Doerr, Michael Gnewuch and Nils Hebbinghaus 

title:  Discrepancy of Products of Hypergraphs 
keywords:  discrepancy, hypergraphs, Ramsey theory 
abstract: 
For a hypergraph
H
= (V,
E
)
d
fold symmetric product is
Δ
. We give several upper and lower bounds for the
d
H
= (V
d
,{ E
d
 E ∈
E
})
c
color discrepancy of such products. In particular,
we show that the bound
disc
(Δ
d
H
,2) ≤
disc
(
H
,2)
d
in [B. Doerr, A. Srivastav, and
P. Wehr, Discrepancy of
C
artesian products of arithmetic progressions,
Electron. J. Combin. 11(2004), Research Paper 5, 16 pp.]
cannot be extended to more than
c = 2
colors. In fact, for any
c
and
d
such that
c
does not divide
d!
, there are hypergraphs having arbitrary large
discrepancy and
disc
(Δ
d
H
,c) = Ω
d
(
disc
(
H
,c)
d
)
c
and
d
), in these cases the symmetric product behaves no
better than the general direct product
H
d
disc
(
H
d
,c) = O
c,d
(
disc
(
H
,c)
d
)

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reference:  Benjamin Doerr and Michael Gnewuch and Nils Hebbinghaus (2005), Discrepancy of Products of Hypergraphs, in 2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05), Stefan Felsner (ed.), Discrete Mathematics and Theoretical Computer Science Proceedings AE, pp. 323328 
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