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finiteseq10_8_98 - 1 LONG FINITE SEQUENCES by Harvey M...

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1 LONG FINITE SEQUENCES by Harvey M. Friedman* Department of Mathematics Ohio State University Columbus, Ohio 43210 [email protected] www.math.ohio-state.edu/~friedman/ October 8, 1998 ABSTRACT Let k be a positive integer. There is a longest finite sequence x 1 ,...,x n in k letters in which no consecutive block x i ,...,x 2i is a subsequence of any other consecutive block x j ,...,x 2j . Let n(k) be this longest length. We prove that n(1) = 3, n(2) = 11, and n(3) is incomprehensibly large. We give a lower bound for n(3) in terms of the familiar Ackerman hierarchy. We also give asymptotic upper and lower bounds for n(k). We view n(3) as a particularly elemental description of an incomprehensibly large integer. Related problems involving binary sequences (two letters) are also addressed. We also report on some recent computer explorations of R. Dougherty which we use to raise the lower bound for n(3). TABLE OF CONTENTS 1. Finiteness, and n(1),n(2). 2. Sequences of fixed length sequences. 3. The Main Lemma. 4. Lower bound for n(3). 5. The function n(k). 6. Related problems and computer explorations. 1. FINITENESS, AND n(1),n(2) We use Z for the set of all integers, Z + for the set of all positive integers, and N for the set of all nonnegative integers. Sequences can be either finite or infinite. For sequences x, it will be convenient to write x[i] for x i , which is the term of x with index i. Unless stated otherwise, all nonempty sequences are indexed starting with 1. Sometimes we consider sequences indexed starting at a positive integer greater than 1.
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2 Let x[1],...,x[n] and y[1],...,y[m] be two finite sequences, where n,m 0. We use the usual notion of subsequence. Thus x[1],...,x[n] is a subsequence of y[1],...,y[m] if and only if there exist 1 £ i 1 < ... < i n £ m such that for all 1 £ j £ n, we have x[j] = y[i j ]. We say that x[1],...,x[n] is a proper subsequence of y[1],...,y[m] if and only if x[1],...,x[n] is a subsequence of y[1],...,y[m] and n < m. The focus of this paper is on finite combinatorics. But we start with the following theorem in infinitary combinatorics. It is a special case of the familiar fundamental result from wqo theory known as Higman°s Lemma [Hi52]. For the sake of completeness, we give the Nash-Williams proof from [Nw63] (adapted to this special case) of the second claim in Theorem 1.1. Note how remarkably nonconstructive this simplest of all proofs is. Let x = x[1],...,x[n] be any sequence. We say that x has property * if and only if for no i < j £ n/2 is it the case that x[i],...,x[2i] is a subsequence of x[j],...,x[2j]. More generally, let x = x[m],...,x[n] be a sequence indexed from m. We say that x has property * if and only if for no m £ i < j £ n/2 is it the case that x[i],...,x[2i] is a subsequence of x[j],...,x[2j]. These definitions are also made for infinite sequences by simply omitting ± £ n/2.² For any set A, let A* be the set of all finite sequences from A (including the empty sequence).
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