Poset Ramsey number $R(P,Q_n)$. I. Complete multipartite posets

TL;DR

This paper investigates the poset Ramsey number involving complete multipartite posets and hypercubes, establishing an upper bound that improves understanding of poset embeddings in colored Boolean lattices.

Contribution

The paper provides a new upper bound for the poset Ramsey number involving complete multipartite posets and hypercubes, advancing the theoretical understanding of poset Ramsey theory.

Findings

Established an upper bound: R(K_{t_1,...,t_ell},Q_n) ≤ n + (2+o(1)) * ell * n / log n

Introduced bounds for poset Ramsey numbers involving multipartite and hypercube posets

Enhanced the theoretical framework for poset embeddings in colored Boolean lattices

Abstract

A poset $(P',\le_{P'})$ contains a copy of some other poset $(P,\le_P)$ if there is an injection $f\colon P'\to P$ where for every $X,Y\in P$, $X\le_P Y$ if and only if $f(X)\le_{P'} f(Y)$. For any posets $P$ and $Q$, the poset Ramsey number $R(P,Q)$ is the smallest integer $N$ such that any blue/red coloring of a Boolean lattice of dimension $N$ contains either a copy of $P$ with all elements blue or a copy of $Q$ with all elements red. We denote by $K_{t_1,\dots,t_\ell}$ a complete $\ell$-partite poset, i.e.\ a poset consisting of $\ell$ pairwise disjoint sets $A^i$ of size $t_i$, $1\le i\le \ell$, such that for any $i,j\in\{1,\dots,\ell\}$ and any two $X\in A^{i}$ and $Y\in A^{j}$, $X<Y$ if and only if $i<j$. In this paper we show that $R(K_{t_1,\dots,t_\ell},Q_n)\le n+\frac{(2+o_n(1))\ell n}{\log n}$.