Ramsey numbers for partially-ordered sets

Paper

Abstract

We say that a poset

contains a copy (resp.~an induced copy) of a poset

if there is an injection

such that for any

if (resp.~if and only if)

. Let

be a family of posets such that

and

for each

. For given

posets

, the \emph{weak (resp.~strong) poset Ramsey number for

-chains} is the smallest number

such that for any coloring of

-chains in

with

colors, say

, there is a monochromatic (resp.~induced) copy of the poset

in color

for some

. In this paper, we give several lower and upper bounds on the weak and strong poset Ramsey number for

-chains.