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 $f: \mathbb{Z} _{\geq0} \rightarrow \mathbb{Z} _{\geq0}$ 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.?
