1
LONG FINITE SEQUENCES
by
Harvey M. Friedman*
Department of Mathematics
Ohio State University
Columbus, Ohio 43210
[email protected]
www.math.ohiostate.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 NashWilliams 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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