Non-dominating sequences of vectors using only resets and increments

TL;DR

This paper introduces a variant of Dickson's lemma involving vector entries that can be reset or incremented, providing a non-dominating sequence matching the known upper bound, advancing understanding of vector sequence behaviors.

Contribution

It presents a new variant of Dickson's lemma with reset and increment operations, and constructs a non-dominating sequence that matches the known upper bound.

Findings

Constructed a non-dominating sequence of length $2^{2^{ heta(n)}}$

Sequence length matches the previously known upper bound

Provides insights into vector sequence behaviors under reset and increment operations

Abstract

We consider a variant of Dickson lemma, where each entry of a vector can be reseted or incremented by 1 in respect to the previous one. We give an example of non dominating sequence of length $2^{2^{\theta (n)}}$. It perfectly match the previously known upperbound.