Abstract
AbstractThe Dushnik–Miller dimension of a poset P is the least d for which P can be embedded into a product of d chains. Lewis and Souza isibility order on the interval of integers $$[N/\kappa , N]$$
[
N
/
κ
,
N
]
is bounded above by $$\kappa (\log \kappa )^{1+o(1)}$$
κ
(
log
κ
)
1
+
o
(
1
)
and below by $$\Omega ((\log \kappa /\log \log \kappa )^2)$$
Ω
(
(
log
κ
/
log
log
κ
)
2
)
. We improve the upper bound to $$O((\log \kappa )^3/(\log \log \kappa )^2).$$
O
(
(
log
κ
)
3
/
(
log
log
κ
)
2
)
.
We deduce this bound from a more general result on posets of multisets ordered by inclusion. We also consider other divisibility orders and give a bound for polynomials ordered by divisibility.
Funder
National Science Foundation
National Security Agency
CYAN Mathematics Undergraduate Activities Fund
Publisher
Springer Science and Business Media LLC
Subject
Computational Theory and Mathematics,Geometry and Topology,Algebra and Number Theory,Discrete Mathematics and Combinatorics
Reference14 articles.
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