Ramsey numbers for partially-ordered sets

TL;DR

This paper investigates the minimal size of posets in a growing family needed to guarantee monochromatic or induced copies of given posets under any coloring of their chains, extending Ramsey theory to partially ordered sets.

Contribution

It provides new lower and upper bounds on weak and strong poset Ramsey numbers for chains in a family of expanding posets.

Findings

Established bounds for weak poset Ramsey numbers.

Established bounds for strong poset Ramsey numbers.

Extended classical Ramsey concepts to partially ordered sets.

Abstract

We say that a poset $Q$ contains a copy (resp.~an induced copy) of a poset $P$ if there is an injection $f : P \to Q$ such that for any $x,y \in P$, $f(x)\leq f(y)$ in $Q$ if (resp.~if and only if) $x\leq y$ in $P$. Let $\mathcal{Q}=\{Q_{n} : n\geq 1\}$ be a family of posets such that $Q_n\subseteq Q_{n+1}$ and $|Q_n|<|Q_{n+1}|$ for each $n$. For given $k$ posets $P_1, P_2, \dots , P_k$, the \emph{weak (resp.~strong) poset Ramsey number for $t$-chains} is the smallest number $n$ such that for any coloring of $t$-chains in $Q_n\in \mathcal{Q}$ with $k$ colors, say $1,2, \dots, k$, there is a monochromatic (resp.~induced) copy of the poset $P_i$ in color $i$ for some $1\leq i\leq k$. In this paper, we give several lower and upper bounds on the weak and strong poset Ramsey number for $t$-chains.