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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 $B$ .

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