1
P
01,
P
00 Incompleteness: finite graph theory
by
Harvey M. Friedman
February 25, 2006
In this abstract, a digraph is a directed graph with no
loops and no multiple edges. Thus all digraphs will be
simple. The results will be the same if we allow loops.
A dag is a directed graph with no cycles.
Let G be a digraph. We write V(G) for the set of all
vertices in G, and E(G) for the set of all edges in G.
Let A V(G). We write GA for the set of all destinations
of edges in G whose origins lie in A. I.e., GA = {y:
(
$
x)((x,y) E(G))}.
We begin by quoting a well known theorem about directed
acyclic graphs, or so called dags. We call it the
complementation theorem, but we have been told that it is
rather ordinary fundamental fare in dag theory.
COMPLEMENTATION THEOREM (finite dags). Let G be a finite
dag. There is a unique set A V(G) such that GA = V(G)\A.
We can look at the Complementation Theorem in terms of a
large independent set. We say that A V(G) is independent
in G if and only if there is no edge connecting any two
elements of A.
COMPLEMENTATION THEOREM (finite dags). Every finite dag has
a unique independent set A such that V(G)\A GA.
A digraph on a set E is a digraph G where V(G) = E.
We will focus on digraphs whose vertex set is of the form
[1,n]
k
. Here k,n
≥
1 and [1,n] = {1,2,...,n}.
An upgraph on [1,n]
k
is a digraph on [1,n]
k
such that for
all (x,y) E(G), max(x) < max(y).
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 Fall '08
 JOSHUA
 Math, Set Theory, Graph Theory, Naive set theory, Axiom of choice, Inaccessible cardinal, Complementation Theorem

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