Poset Ramsey Number $R(P,Q_n)$. II. Antichains

Abstract

For two posets $(P,{\le }_P)$ and $(P^{\prime },{\le }_{P^{\prime }})$, we say that $P^{\prime }$ contains a copy of $P$ if there exists an injective function $f:P^{\prime }\to P$ such that for every two $X,Y\in P$, $X{\le }_{P}Y$ if and only if $f(X){\le }_{P^{\prime }}f(Y)$. Given two posets $P$ and $Q$, let the poset Ramsey number $R(P,Q)$ be the smallest integer $N$ such that any coloring of the elements of an $N$-dimensional Boolean lattice in blue or red contains either a copy of $P$ where all elements are blue or a copy of $Q$ where all elements are red. We determine the poset Ramsey number $R(A_t,Q_n)$ of an antichain versus a Boolean lattice for small $t$ by showing that $R(A_t,Q_n)=n+3$ for $3\le t\le \log \log n$.

arXiv commentMerged with arXiv:2303.04462