Duality in the Combinatorics of Posets
| Speaker: | William T. Trotter |
|---|---|
| Affiliation: | Georgia Institute of Technology |
| Room: | Mathematics 3 (M3) 2134 |
Abstract:
Every
10
years
or
so,
there
have
been
two
closely
connected
theorems
in
the
combinatoricst
of
posets,
one
for
chains
and
one
for
antichains.
Typically,
the
statements
are
exactly
the
same
when
roles
are
reversed,
but
the
proofs
are
markedly
different.
And
on
rare
occasions,
there
has
been
a
case
where
there
is
a
natural
candidate
for
a
dual
statement
but
it
turned
out
not
to
be
true.
The
classic
pair
of
theorems
due
to
Dilworth
and
Mirsky
were
the
starting
point
for
this
pattern,
followed
by
the
more
general
pair
known
respectively
as
the
Greene-Kleitman
and
Greene
theorems
dealing
with
saturated
partitions.
More
recently,
we
have
the
pair
of
theorems
due
to
Duffus-Sands
and
Howard-Trotter
dealing
with
pairwise
disjoint
families
of
subsets.
And
within
the
last
month,
a
new
pair
has
been
discovered
dealing
with
matchings
in
the
comparability
and
incomparability
graphs
of
a
poset.