3 problems

Milner–Sauer conjecture. If $\operatorname{cf}(P,\le)=\lambda>\operatorname{cf}(\lambda)=\kappa$, then $P$ contains an antichain of size $\kappa$.

For a positive integer $k$, let $\mathbf{k}+\mathbf{k}$ be the poset consisting of two disjoint $k$-element chains with no comparabilities between distinct chains. Let…

For positive integers $k$ and $w$, let $f(k,w)$ be the maximum size of a subset of $\mathbb{Z}^w$ in which every two vectors are $1$-crossing but no two vectors are $k$-crossing. F…