# On subsets of $\mathbb N^2$ with elements not comparable w.r.t. componentwise order

Let $$\mathbb N$$ denote the set of nonnegative integers . For $$(a,b);(c,d)\in \mathbb N^2$$, define $$(a,b)\le (c,d)$$ iff $$a\le c$$ and $$b\le d$$. Let us call call a subset $$S\subseteq \mathbb N^2$$ to be an antichain if no two elements of $$S$$ are comparable by $$\le$$. Let us call an antichain to be a maximal antichain if it is a maximal one, w.r.t. to inclusion, among all antichains.

For a subset $$A\subseteq \mathbb N^2$$, define $$A_n:=A+\{(j,n-j):0\le j\le n\}=\{(a+j,b+n-j):(a,b)\in A, 0\le j\le n\}$$.

My question is: Let $$B$$ be a maximal antichain such that for some $$a,b\ge 1$$, we have $$(a,0);(0,b)\in B$$ . Then how to find an integer $$n\ge 1$$ (depending on elements of $$B$$) such that $$B_m$$ is a maximal antichain for all $$m\ge n$$ ?

Specifically, if $$n_B$$ is the smallest integer such that $$B_m$$ is a maximal antichain $$\forall m\ge n_B$$, then I'm looking for an upper bound for $$n_B$$ in terms of properties of $$B$$ .