Erd\H{o}s-Hajnal problems for posets

TL;DR

This paper investigates the Erd ext{"o}s-Hajnal problem for posets, establishing bounds on the poset Erd ext{"o}s-Hajnal number and improving lower bounds for the poset Ramsey number for Boolean lattices.

Contribution

It introduces bounds on the poset Erd ext{"o}s-Hajnal number for various posets and improves the lower bound for the poset Ramsey number of Boolean lattices.

Findings

Bounds on the poset Erd ext{"o}s-Hajnal number for general and specific posets.

Improved lower bound for the poset Ramsey number $R(Q_n,Q_n)$ to greater than 2.02n.

Demonstrated that $R(Q_n,Q_n)$ exceeds the previous lower bound of 2n+1.

Abstract

We say that a poset $(Q,\le_{Q})$ contains an induced copy of a poset $(P,\le_P)$ if there is an injective function $\phi\colon P\to Q$ such that for every two $X,Y\in P$,\;\;$X\le_P Y$ if and only if $\phi(X)\le_Q \phi(Y)$. We denote the Boolean lattice $(2^{[n]},\subseteq)$ by $Q_n$. Given a fixed $2$-coloring $c$ of a poset $P$, the poset Erd\H{o}s-Hajnal number of this colored poset is the smallest integer $N$ such that every $2$-coloring of the Boolean lattice $Q_N$ contains an induced copy of $P$ colored as in $c$, or a monochromatic induced copy of $Q_n$. We present bounds on the poset Erd\H{o}s-Hajnal number of general colored posets, antichains, chains, and small Boolean lattices. Let the poset Ramsey number $R(Q_n,Q_n)$ be the least $N$ such that every $2$-coloring of $Q_N$ contains a monochromatic induced copy of $Q_n$. As a corollary, we show that $R(Q_n,Q_n)> 2.02n$, improving on the best known lower bound $2n+1$ by Cox and Stolee \cite{CS}.