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

TL;DR

This paper determines the poset Ramsey number for an antichain versus a Boolean lattice, showing it equals n+3 for small t, specifically when 3 ≤ t ≤ log log n, advancing understanding of poset embeddings in Boolean lattices.

Contribution

It establishes the exact value of the poset Ramsey number R(A_t,Q_n) for small t, specifically for t between 3 and log log n, for the first time.

Findings

R(A_t,Q_n) = n + 3 for 3 ≤ t ≤ log log n

Provides bounds for poset Ramsey numbers involving antichains and Boolean lattices

Advances knowledge of poset embeddings in combinatorics

Abstract

For two posets $(P,\le_P)$ and $(P',\le_{P'})$, we say that $P'$ contains a copy of $P$ if there exists an injective function $f\colon P'\to P$ such that for every two $X,Y\in P$, $X\le_P Y$ if and only if $f(X)\le_{P'} f(Y)$. 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 $3\le t\le \log \log n$.