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1 FINITE TREES AND THE NECESSARY USE OF LARGE CARDINALS by Harvey M. Friedman Department of Mathematics Ohio State University [email protected] www.math.ohio-state.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 nec-essary 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 k-tree is a finite rooted tree whose root is and whose children are k element subsets of N. The trivial k-tree is the k-tree 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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2 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 k-tree 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 k-tree 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 k-trees T 0 ... T n as follows. T 0 is the trivial k-tree. 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 k-tree with any finite subset of [N] k as its children.
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