Poset Ramsey number $R(P,Q_n)$. II. N-shaped poset

TL;DR

This paper introduces a new method using blockers to bound the poset Ramsey number R(P, Q_n), and determines its asymptotic behavior for a specific four-element poset, advancing understanding of poset Ramsey theory.

Contribution

It develops a duality-based approach with blockers to establish upper bounds on R(P, Q_n) and computes its asymptotic value for a particular four-element poset.

Findings

R(, Q_n) = n + (n / \u221a{ ext{log} n}) asymptotically

Introduces a duality between copies of Q_n and blockers in poset Ramsey theory

Provides new bounds and methods for analyzing poset Ramsey numbers

Abstract

Given partially ordered sets (posets) $(P, \leq_P)$ and $(P', \leq_{P'})$, we say that $P'$ contains a copy of $P$ if for some injective function $f\colon P\rightarrow P'$ and for any $A, B\in P$, $A\leq _P B$ if and only if $f(A)\leq_{P'} f(B)$. For any posets $P$ and $Q$, the poset Ramsey number $R(P,Q)$ is the least positive integer $N$ such that no matter how the elements of an $N$-dimensional Boolean lattice are colored in blue and red, there is either a copy of $P$ with all blue elements or a copy of $Q$ with all red elements. We focus on the poset Ramsey number $R(P, Q_n)$ for a fixed poset $P$ and an $n$-dimensional Boolean lattice $Q_n$, as $n$ grows large. It is known that $n+c_1(P) \leq R(P,Q_n) \leq c_2(P) n$, for positive constants $c_1$ and $c_2$. However, there is no poset $P$ known, for which $R(P, Q_n)> (1+\epsilon)n$, for $\epsilon >0$. This paper is devoted to a new method for finding upper bounds on $R(P, Q_n)$ using a duality between copies of $Q_n$ and sets of elements that cover them, referred to as blockers. We prove several properties of blockers and their direct relation to the Ramsey numbers. Using these properties we show that $R(\mathcal{N},Q_n)=n+\Theta(n/\log n)$, for a poset $\mathcal{N}$ with four elements $A, B, C, $ and $D$, such that $A<C$, $B<D$, $B<C$, and the remaining pairs of elements are incomparable.