Erdős-Hajnal problems for posets

TL;DR

This work extends Erdős–Hajnal-type Ramsey questions to posets by defining the poset Erdős-Hajnal number $ ilde{R}(\dot P,Q_n)$ for fixed colored posets. It develops a general framework distinguishing diverse and non-diverse colored posets, proving linear-in-$n$ bounds for many fixed colored posets and, in particular, tight results for antichains and chains. A key methodological contribution is a chain-reduction technique via alternating red/blue subchains and phased analysis of $Q_N$, enabling linear upper bounds and precise constants in several cases. As a corollary, the paper improves the known lower bound for the poset Ramsey number $R(Q_n,Q_n)$ to $>2.02n$, advancing the understanding of monochromatic versus heterochromatic substructure thresholds in Boolean lattices.

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 $φ\colon P\to Q$ such that for every two $X,Y\in P$,\;\;$X\le_P Y$ if and only if $φ(X)\le_Q φ(Y)$. We denote the Boolean lattice $(2^{[n]},\subseteq)$ by $Q_n$. Given a fixed $2$-coloring $c$ of a poset $P$, the poset Erdős-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 Erdős-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}.

Figures (8)

• Figure 1: Alternating chains and non-monochromatic colorings of $Q_2$.
• Figure 2: A colored chain $\dot C$ and a $\dot C$-free blue/red coloring of $Q_4$ with sets $\mathcal{F}^{\dot C}_{i}$, $i\in[4]$.
• Figure 3: A colored chain $\dot C$, families $\mathcal{F}_i$ in $\mathcal{Q}$, $\mathcal{F}'_i$ in $\mathcal{Q}'$, and $\mathcal{H}_j$ partitioning $\mathcal{Q}'$, where $t=9$, $s=5$, and $\lambda=5$.
• Figure 4: Blue/red coloring of $\mathcal{Q}([N])$ based on $\mathcal{S}$ and $\mathcal{T}$ in Construction \ref{['constr:EHchain']}.
• Figure 5: Examples for families $\mathcal{K}_s(A,B)$, $\mathcal{K}_t(A,B)$, and $\mathcal{N}_t(S)$.

Theorems & Definitions (30)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Corollary 6
• Theorem 7
• Lemma 8: CCLL
• Corollary 9: AW
• Lemma 10: AW