Ramsey numbers for partially-ordered sets

TL;DR

This paper extends the concept of Ramsey numbers to graphs with vertices partially ordered, exploring their properties and differences from ordered Ramsey numbers, especially in the context of the Boolean lattice.

Contribution

It formalizes the use of partially-ordered sets in Ramsey problems and investigates their connections to Turán-type problems, highlighting differences with ordered Ramsey numbers.

Findings

Significant differences between Ramsey numbers on Boolean lattices and ordered Ramsey numbers.

Large antichains in partial orders influence Ramsey number behavior.

Connections established between Ramsey theory and Turán-type problems in posets.

Abstract

We present a refinement of Ramsey numbers by considering graphs with a partial ordering on their vertices. This is a natural extension of the ordered Ramsey numbers. We formalize situations in which we can use arbitrary families of partially-ordered sets to form host graphs for Ramsey problems. We explore connections to well studied Tur\'an-type problems in partially-ordered sets, particularly those in the Boolean lattice. We find a strong difference between Ramsey numbers on the Boolean lattice and ordered Ramsey numbers when the partial ordering on the graphs have large antichains.