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FINITE TREES AND THE NECESSARY USE OF LARGE CARDINALS
by
Harvey M. Friedman
Department of Mathematics
Ohio State University
[email protected]
www.math.ohiostate.edu/~friedman/
January 10, 1998
We introduce the basic concept of insertion rule that speci
fies the placement of new vertices into finite trees. We
prove that every “good” insertion rule generates a tree with
simple structural properties in the style of classical Ramsey
theory. This is proved using large cardinal axioms going well
beyond the usual axioms for mathematics.
And this result cannot be proved without these large cardinal
axioms. We also prove that every insertion rule greedily
generates a tree with these same structual properties. It is
also necessary and sufficient to use the same large
cardinals. The results suggest a new area of research 
"greedy Ramsey theory."
A partial ordering is a pair (X,
£
), where X is a nonempty
set, and
£
is reflexive, transitive, and antisymmetric. The
ancestors of x in X are just the y < x. A tree T = (V,
£
) is a
partial ordering with a least element (root), where the the
ancestors of any x in V form a finite linearly ordered set
under ≤.
If x < y and for no z is x < z < y, then we say that y is a
child of x and x is the parent of y. V = V(T) is the set of
vertices of the tree T = (V,
£
). Every vertex other than the
root has a unique parent; i.e., is a child.
For finite rooted trees, T
1
T
2
means that T
1
,T
2
have the
same root, and if x is the parent of y in T
1
, then x is the
parent of y in T
2
. I.e., no parent/child bond is broken by
going from T
1
to T
2
.
Let N = {1,2,.
..}. A ktree is a finite rooted tree whose
root is and whose children are k element subsets of N.
The trivial ktree is the ktree with the single vertex .
We write [N]
k
be the set of all k element subsets N and [N]
k+
for [N]
k
{ }. We use
£
* for the standard reverse lexico
graphic ordering on [N]
k
, in which sets are compared accor
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ding to the lexicographic ordering on N
k
after the sets are
placed in strictly descending order. We extend this linear
ordering to [N]
k+
by placing at the top.
Let T be a ktree and x [N]
k
. We say that x dominates T if
and only if for all y Ch(T), we have x >* y.
Insertion rules specify the placement of new, dominating,
vertices into trees.
Formally, an insertion rule on TR(k) is a function f defined
on exactly the pairs (T,x), where T is a ktree and x [N]
k
dominates T, where each defined f(T,x) lies in V(T).
How do we use an insertion rule on TR(k)? Let A [N]
k
be
finite and write A = v
1
<* .
.. <* v
n
. Construct ktrees T
0
... T
n
as follows. T
0
is the trivial ktree. T
i+1
is the k
tree obtained by inserting v
i+1
into T
i
as a new child of the
vertex f(T
i
,v
i+1
). We write T
i+1
= T
i
/v
i+1
,f(T
i
,v
i+1
). Clearly
Ch(T
n
) = A.
In this way, f generates a unique ktree with any finite
subset of [N]
k
as its children.
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 Fall '08
 JOSHUA
 Math, Set Theory, Tn, ZFC, insertion rule

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