305 – A brief update on n(3)

April 12, 2012

This continues the previous post on A lower bound for n(3).

Only recently I was made aware of a note dated November 22, 2001, posted on Harvey Friedman‘s page, “Lecture notes on enormous integers”. In section 8, Friedman recalls the definition of the function n(k), the length of the longest possible sequence x_1,x_2,\dots,x_n from \{1,2,\dots,k\} with the property that for no i<j\le n/2, the sequence x_i,x_{i+1},\dots,x_{2i} is a subsequence of x_j,x_{j+1},\dots,x_{2j}.

Friedman says that “A good upper bound for n(3) is work in progress”, and states (without proof):

Theorem. n(3)\le A_k(k), where k=A_5(5).

Here, A_1,A_2,\dots are the functions of the Ackermann hierarchy (as defined in the previous post).

He also indicates a much larger lower bound for n(4). We need some notation first: Let A(m)=A_m(m). Use exponential notation to denote composition, so A^3(n)=A(A(A(n))).

Theorem. Let m=A(187196). Then n(4)>A^m(1).

He also states a result relating the rate of growth of the function n(\cdot) to what logicians call subsystems of first-order arithmetic. A good reference for this topic is the book Metamathematics of First-order Arithmetic, by Hájek and Pudlák, available through Project Euclid.

There is a recent question on MathOverflow on this general topic.