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Pi01Incomp022506

# Pi01Incomp022506 - 1 P01,P00 Incompleteness finite graph...

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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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