MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).

First, we define a sequence $t_{1},t_{2},\cdots,t_{k}$ of n-tuples dicksonian, if $\forall 1\leq i < j\leq k,$ there does not exist a non-negative n-tuple t such that $t_{i}+t=t_{j}.$ For example, any lexicographically decreasing sequnence is dicksonian. By Dickson's lemma, every dicksonian sequence is finite. Let $(a_{1}^{1},\cdots,a_{n}^{1}),(a_{1}^{2},\cdots,a_{n}^{2}),\cdots,(a_{1}^{k},\cdots,a_{n}^{k})$ be a dicksonian sequence of n-tuples of non-negative integers such that $\sum_{i=1}^{n}(a_{i}^{j})=f(j)$ for all $j,1\leq j\leq k,$ where $f:\mathbb{Z}{\geq0}\rightarrow f: \mathbb{Z} {\geq0}$ _{\geq0} \rightarrow \mathbb{Z} _{\geq0}$is a fixed function. Note that in the paper, "G. Moreno Socias, An Ackermannian polynomial ideal" it actually considered the maximal length of a dicksonian sequence such that$f(1)=d,f(i+1)=f(i)+1,\forall i\geq 1$, and this result is represented as a Ackermann function. Considering the characteristic of the dicksonian sequence satisfying the requirement with the maximal length with$n=3,d=3,$given at the end of this paper, I want to ask the following question: What is the possible maximal length for a dickson sequence such that$f(1)=d,f(i+1)=f(i)+1,\forall i\geq 1$, and the sum of the first two entries of every n-tuple in this dicksonian sequence is a fixed number, say m.? Note that the position of the two entries with a fixed sum in a n-tuple may further affect the final result, I may further ask the following question: What is the possible maximal length for a dickson sequence such that$f(1)=d,f(i+1)=f(i)+1,\forall i\geq 1$, and the sum of the two entries at position$i_0,j_0, 1\leq i_0\lt j_0\leq n$of every n-tuple in this dicksonian sequence is a fixed number, say m.? 3 added 1 characters in body; added 2 characters in body First, we define a sequence$t_{1},t_{2},\cdots,t_{k}$of n-tuples dicksonian, if$\forall 1\leq i < j\leq k,$there does not exist a non-negative n-tuple t such that$t_{i}+t=t_{j}.$For example, any lexicographically decreasing sequnence is dicksonian. By Dickson's lemma, every dicksonian sequence is finite. Let$(a_{1}^{1},\cdots,a_{n}^{1}),(a_{1}^{2},\cdots,a_{n}^{2}),\cdots,(a_{1}^{k},\cdots,a_{n}^{k})$be a dicksonian sequence of n-tuples of non-negative integers such that$\sum_{i=1}^{n}(a_{i}^{j})=f(j)$for all$1\leq i{\geq0}\rightarrow j,1\leq j\leq k,$where$f:\mathbb{Z}{\geq0}\rightarrow \mathbb{Z}{\geq0}$mathbb{Z}{\geq0}$ is a fixed function.

Note that in the paper, "G. Moreno Socias, An Ackermannian polynomial ideal" it actually considered the maximal length of a dicksonian sequence such that $f(1)=d,f(i+1)=f(i)+1,\forall i\geq 1$, and this result is represented as a Ackermann function. Considering the characteristic of the dicksonian sequence satisfying the requirement with the maximal length with $n=3,d=3,$ given at the end of this paper, I want to ask the following question:

What is the possible maximal length for a dickson sequence such that $f(1)=d,f(i+1)=f(i)+1,\forall i\geq 1$, and the sum of the first two entries of every n-tuple in this dicksonian sequence is a fixed number, say m.?

Note that the position of the two entries with a fixed sum in a n-tuple may further affect the final result, I may further ask the following question:

What is the possible maximal length for a dickson sequence such that $f(1)=d,f(i+1)=f(i)+1,\forall i\geq 1$, and the sum of the two entries at position $i_0,j_0, 1\leq i_0\lt j_0\leq n$ of every n-tuple in this dicksonian sequence is a fixed number, say m.?

2 changed "<" to "\lt "

First, we define a sequence $t_{1},t_{2},\cdots,t_{k}$ of n-tuples dicksonian, if for all $1\leq i{\geq0}\rightarrow \mathbb{Z}{\geq0}$ is a fixed function.

Note that in the paper, "G. Moreno Socias, An Ackermannian polynomial ideal" it actually considered the maximal length of a dicksonian sequence such that $f(1)=d,f(i+1)=f(i)+1,\forall i\geq 1$, and this result is represented as a Ackermann function. Considering the characteristic of the dicksonian sequence satisfying the requirement with the maximal length with $n=3,d=3,$ given at the end of this paper, I want to ask the following question:

What is the possible maximal length for a dickson sequence such that $f(1)=d,f(i+1)=f(i)+1,\forall i\geq 1$, and the sum of the first two entries of every n-tuple in this dicksonian sequence is a fixed number, say m.?

Note that the position of the two entries with a fixed sum in a n-tuple may further affect the final result, I may further ask the following question:

What is the possible maximal length for a dickson sequence such that $f(1)=d,f(i+1)=f(i)+1,\forall i\geq 1$, and the sum of the two entries at position $i_0,j_0, 1\leq i_0 i_0\lt j_0\leq n$ of every n-tuple in this dicksonian sequence is a fixed number, say m.?

1