% EJC papers *must* begin with the following two lines. 
\documentclass[12pt]{article}
\usepackage{e-jc}
\specs{P4.36}{24(4)}{2017}


% Please remove all other commands that change parameters such as
% margins or pagesizes.

% only use standard LaTeX packages
% only include packages that you actually need

% we recommend these ams packages
\usepackage{amsthm,amsmath,amssymb}

% we recommend the graphicx package for importing figures
\usepackage{graphicx}

% use this command to create hyperlinks (optional and recommended)
\usepackage[colorlinks=true,citecolor=black,linkcolor=black,urlcolor=blue]{hyperref}

% use these commands for typesetting doi and arXiv references in the bibliography
\newcommand{\doi}[1]{\href{http://dx.doi.org/#1}{\texttt{doi:#1}}}
\newcommand{\arxiv}[1]{\href{http://arxiv.org/abs/#1}{\texttt{arXiv:#1}}}

% all overfull boxes must be fixed; 
% i.e. there must be no text protruding into the margins


% declare theorem-like environments
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{fact}[theorem]{Fact}
\newtheorem{observation}[theorem]{Observation}
\newtheorem{claim}[theorem]{Claim}

\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{open}[theorem]{Open Problem}
\newtheorem{problem}[theorem]{Problem}
\newtheorem{question}[theorem]{Question}

\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{note}[theorem]{Note}

\newcommand{\xx}{{\mathcal{X}}}
\newcommand{\yy}{{\mathcal{Y}}}
\newcommand{\zz}{\mathcal{Z}}
\renewcommand{\aa}{\mathcal{A}}
\newcommand{\bb}{\mathcal{B}}
\newcommand{\cc}{\mathcal{C}}
\newcommand{\lr}{\Leftrightarrow}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% if needed include a line break (\\) at an appropriate place in the title

\title{\bf Note on a Ramsey theorem \\ for posets with linear extensions}

% input author, affilliation, address and support information as follows;
% the address should include the country, and does not have to include
% the street address 

\author{Andrii Arman\\
\small Department of Mathematics\\[-0.8ex]
\small University of Manitoba\\[-0.8ex] 
\small Winnipeg, Canada\\
\small\tt armana@myumanitoba.ca\\
\and
Vojt\v{e}ch R\"{o}dl\thanks{Supported by NSF grant DMS 1301698}\\
\small Department of Mathematics and Computer Sciences\\[-0.8ex]
\small Emory University\\[-0.8ex]
\small Atlanta, U.S.A.\\
\small\tt rodl@mathcs.emory.edu
}


% \date{\dateline{submission date}{acceptance date}\\
% \small Mathematics Subject Classifications: comma separated list of
% MSC codes available from http://www.ams.org/mathscinet/freeTools.html}

\date{\dateline{Aug 19, 2016}{Nov 17, 2017}{Dec 8, 2017}\\
\small Mathematics Subject Classifications: 05C55,  06A07}

\begin{document}

\maketitle

% E-JC papers must include an abstract. The abstract should consist of a
% succinct statement of background followed by a listing of the
% principal new results that are to be found in the paper. The abstract
% should be informative, clear, and as complete as possible. Phrases
% like "we investigate..." or "we study..." should be kept to a minimum
% in favor of "we prove that..."  or "we show that...".  Do not
% include equation numbers, unexpanded citations (such as "[23]"), or
% any other references to things in the paper that are not defined in
% the abstract. The abstract will be distributed without the rest of the
% paper so it must be entirely self-contained.

\begin{abstract}
  In this note we consider a Ramsey type result for partially ordered sets. In particular, we give an alternative short proof of a theorem for a posets with multiple linear extensions recently obtained by Solecki and Zhao (2017).

  % keywords are optional
  \bigskip\noindent \textbf{Keywords:} Ramsey theorem; posets
\end{abstract}

\section{Preliminary definitions}
A poset is a pair $(X,P^{X})$, where $X$ is a set and  $P^{X}$ is a partial order on $X$. We consider partial orders that are strict, i.e. not reflexive. 

We say that a  partial order $L^{X}$ on $X$ \textit{extends} a partial order $P^{X}$ on $X$ if for all $x, y \in X$
$$x P^{X} y \Rightarrow x L^{X} y.$$ 

If $(X,P^{X})$ is a poset and $U \subset X$ we denote by $P^{X}|_{U}$ the \textit{restriction} of $P^{X}$ onto $U$. 

Below, we consider collections  $\mathcal{L}^{X}_k=(L_1^{X}, L_2^{X}, \dots, L_k^{X})$, where each of $L_{i}^{X}$ is a linear order on $X$.

\begin{definition}
We denote by $PL^{(k)}$ the set consisting of all
triplets $(X, P^{X}, \mathcal{L}^{X}_k)$, where $(X,P^{X})$ is a poset and each $L_{i}^{X}$ for $ i \in [k]$ is a linear order that extends $P^{X}$.
\end{definition}

\begin{definition}
Let $\xx, \yy \in PL^{(k)}$, where $\xx=(X, P^{X}, \mathcal{L}^{X}_k)$ and $\yy=(Y, P^{Y}, \mathcal{L}^{Y}_k)$.
We  write $\xx \subseteq \yy$ if 
\begin{itemize}
\item $X\subseteq Y$ and $P^{Y}|_X$ extends $P^{X}$.
\item $L^{Y}_{i}|_X=L^{X}_{i}$ for all $i\in [k]$.
\end{itemize} 
\end{definition}


\begin{definition}

Let $\xx, \yy \in PL^{(k)}$, where $\xx=(X, P^{X}, \mathcal{L}^{X}_k)$ and $\yy=(Y, P^{Y}, \mathcal{L}^{Y}_k)$. We say that a mapping $\pi : X \to Y$ is order preserving for $\xx$ and $\yy$ if  for any $i \in [k]$ and any $x,y \in X$ we have 
$$ x L_{i}^X y \lr \pi(x)L_{i}^{Y}\pi(y) \; \; \;\text{and} \; \; \;  x P^{X}y \lr \pi(x)P^{Y}\pi(y).$$
\end{definition}
\begin{definition} \label{def:isomor}
We say that $\pi$ is an isomorphism between $\xx \in PL^{(k)}$ and $\tilde{\xx} \in PL^{(k)}$ if it is order preserving bijection. We say that $\xx \in PL^{(k)}$ is isomorphic to $\tilde{\xx} \in PL^{(k)}$ if there is an isomorphism between $\xx$ and $\tilde{\xx}$.
\end{definition}

\begin{definition}\label{def:copy}
Let  $k > 0$ and  $\xx, \yy \in PL^{(k)}$. We say that $\tilde{\xx} \in PL^{(k)}$ is a copy of $\xx$ in $\yy$ if 
$\tilde{\xx} \subseteq \yy$ and $\tilde{\xx}$ is isomorphic to $\xx$.
For $\xx, \yy \in PL^{(k)}$ denote by  
$\binom{\yy}{\xx}$ the set of all copies of $\xx$ in $\yy$.

\end{definition}
For any $\tilde{\xx} \in \binom{\yy}{\xx}$ there is unique order preserving mapping $\pi: X \to \tilde{X}$. On other hand, any order preserving mapping $\pi : X \to Y$ induces a copy $\tilde{\xx}=\pi(\xx) \in \binom{\yy}{\xx}$.
We identify each $\tilde{\xx} \in \binom{\yy}{\xx}$ with corresponding order preserving mapping $\pi$ and will say that $\pi$ is a copy of $\xx$ in $\yy$ instead of saying that $\tilde{\xx}$ is a copy of $\xx$ in $\yy$ with corresponding order preserving mapping $\pi$. 

The following theorem follows from the result of \cite{NR} (see \cite{Nesetril} and \cite{NR2}). Different proof of Theorem~\ref{thm:one_ext} was also given by Soki\'c~\cite{Sokic1} (using results of \cite{PTW} and \cite{Fouche}). 

\begin{theorem}\label{thm:one_ext}
For any integer $r$ and any $\xx, \yy \in PL^{(1)} $ there is $\zz \in PL^{(1)}$, such that for any $r$-colouring of set $\binom{\zz}{\xx}$ there is $\tilde{\yy}$, a copy of $\yy$ in $\zz$, such that $\binom{\tilde{\yy}}{\xx}$ is monochromatic.  
\end{theorem}

%Ramsey properties of the class of partially ordered sets were considered in \cite{NR} and \cite{PTW}, where all partially ordered sets with P-Ramsey properties were characterised (see also \cite{NR2}). Subsequently some extensions and related results were obtained in \cite{Promel} and \cite{Fouche}, using different methods.  

%Ramsey properties of the class of partially ordered sets were considered in \cite{NR} and \cite{PTW}, where all partially ordered sets with P-Ramsey properties were characterised (see also \cite{NR2}). Subsequently some extensions and related results were obtained in \cite{Promel} and \cite{Fouche}, using different methods.  


Next theorem is a product version of the Theorem \ref{thm:one_ext}, that we are going to use in Section \ref{sec:proof}. Proof of this theorem is based on a standard folkloristic product argument. For similar results of this type see e.g. \cite{Promel}.


\begin{theorem}\label{thm:product}
For any $\xx_i, \yy_{i}\in PL^{(1)}$ with $i\in [k]$  there are $\zz_i \in PL^{(1)}$ with $i \in [k]$, such that for any 2-colouring of set $\binom{\zz_1}{\xx_1} \times \dots \times \binom{\zz_k}{\xx_k}$  there are $\tilde{\yy_i}$, a copies of $\yy_i$ in $\zz_i$ for $i \in [k]$, such that $\binom{\tilde{\yy_1}}{\xx_1} \times \dots \times \binom{\tilde{\yy_k}}{\xx_k}$ is monochromatic.  
\end{theorem}


Based on Theorem \ref{thm:product}, in Section \ref{sec:proof} we provide a proof of the following result, first obtained in \cite{SZ}.

\begin{theorem} \label{thm:main}
For any integer $k$ any $\mathcal{A}, \bb \in PL^{(k)} $ there is $\cc \in PL^{(k)}$, such that for any colouring $2$-colouring of set $\binom{\cc}{\mathcal{A}}$ there is $\tilde{\bb}$, a copy of $\bb$ in $\cc$, such that $\binom{\tilde{\bb}}{\mathcal{A}}$ is monochromatic.  
\end{theorem}

To distinguish between the objects of $PL^{(1)}$, which  play a special role in our proof, and  $PL^{(k)}$ for $k\geq 2$, from now on, we use letters $\xx$, $\yy$ and $\zz$ for elements of $PL^{(1)}$ and $\aa$, $\bb$, $\cc$ for elements of $PL^{(k)}$.

For the ease of notation we will give a proof of Theorem \ref{thm:main} for case $k=2$. The proof of the general case follows the same lines (and is accessible on \href{https://arxiv.org/pdf/1608.05290.pdf}{arxiv.org}).
\section{Properties of join and canonical copies}

First, we define the join of two elements of $PL^{(1)}$.


\begin{definition}\label{def:join}
Let $\zz_{i}=(Z_i, P^{Z_i}, L^{Z_i})\in PL^{(1)}$ for $i =1,2$. Define $\cc=\zz_1 \sqcup \zz_2$ by 
$$\cc=(Z_1 \times Z_2, <_\cc, <_{lx_1}, <_{lx_2}),$$

where  $Z_1 \times Z_2$ is Cartesian product of sets $Z_1$ and $Z_2$, $<_\cc$ is a partial order and $<_{lx_1}, <_{lx_2} $ are linear orders on $Z_1 \times Z_2$ defined by:
$$(x_1,x_2)<_\cc(y_1, y_2) \lr x_1P^{Z_1}y_1 \; \; \text{and} \; \; x_2P^{Z_2}y_2 \;, $$
$$(x_1,x_2)<_{lx_1}(y_1, y_2) \lr x_1L^{Z_1}y_1 \; \; \text{or} \; \; \{ x_1=y_1 \; \text{and} \;  x_{2}L^{Z_{2}}y_{2} \; \}, $$
$$(x_1,x_2)<_{lx_2}(y_1, y_2) \lr x_2L^{Z_2}y_2 \; \; \text{or} \; \; \{ x_2=y_2 \; \text{and} \;  x_{1}L^{Z_{1}}y_{1} \; \}. $$
We say that $\zz_1 \sqcup \zz_2$ is the join of $\zz_1$ and $\zz_2$. 
\end{definition}
Note, that for $\zz_1, \zz_2 \in PL^{(1)}$ we have that $\zz_{1}\sqcup \zz_{2} \in PL^{(2)}$. Indeed, since both $L^{Z_{i}}$ extend $P^{Z_i}$ we infer that both $<_{lx_i}$ also extend $<_\cc$ for $i=1,2$.

\begin{claim}\label{claim:1}
Let $\zz_{i}=(Z_i, P^{Z_i}, L^{Z_i})\in PL^{(1)}$ for $i =1,2 $ and let $\bb=(Y, P^{Y}, {L}^{Y}_{1}, L^{Y}_2) \in PL^{(2)}$. Set $\cc=\zz_1 \sqcup \zz_2$ and let $\pi_{i} : Y \to Z_{i}$ be a copy of $\yy_{i}=(Y, P^{Y}, L_{i}^{Y})$ in $\zz_{i}$ for $i =1,2$. Then the image of the mapping $\pi : Y \to Z_1 \times Z_2$, defined by 
$$\pi(y)=(\pi_1(y), \pi_2(y))$$
for each $y \in Y$, is a copy of $\mathcal{B}$ in $\binom{\cc}{\mathcal{B}}$.
\end{claim}

\begin{remark}\label{remark:1}$ $

\begin{itemize}

\item We say that the image of the mapping $\pi$ from Claim \ref{claim:1}, is a \textit{canonical} copy of $\mathcal{\bb}$ in $\cc=\zz_1 \sqcup \zz_2$.
\item By $\binom{\cc}{\mathcal{B}}_{can} \subseteq \binom{\cc}{\mathcal{B}}$ we denote a set of all canonical copies of $\mathcal{B}$ in $\cc$.
\end{itemize}

\end{remark}

\begin{proof}
We need to verify that $\pi: Y \to Z_1 \times Z_2$ is order preserving for $\aa$ and $\cc$.  
Indeed, we observe that if $x,y \in Y$, then fact that $\pi_i: Y \to Z_{i}$ preserves $P^{Y}$ for $i =1,2$ combined with definition of $\zz_1 \sqcup \zz_2$  yields
$$xP^Yy \lr 
\begin{array}{l}
\pi_1(x)  P^{Z_1} \pi_1(y),\\
\pi_2(x) P^{Z_2} \pi_2(y)
\end{array}
\lr \pi(x)<_\cc \pi(y).$$
Since $\pi_i$ preserves $L^{Y}_i$ for $i=1,2 \;$, we have 
$$ xL_i^Yy \lr \pi_i(x)L_i^{Z_i}\pi_i(y) \lr \pi(x)<_{lx_{i}}\pi(y)$$
for $i =1,2$. Hence, $\pi$ preserves $P^{Y}$ and $L^{Y}_i$ for $i =1,2$.
\end{proof}


For the rest of this section we assume that 
$\cc=\zz_1 \sqcup \zz_2=(Z_1 \times Z_2, <_\cc, <_{lx_1}, <_{lx_2})$,  
$\aa=(X, P^{X}, L_1^{X}, L_2^{X})$ and $\bb=(Y, P^{Y}, L_1^{Y}, L_2^{Y})$.

\begin{fact}\label{fact:lambda}
By construction, $\binom{\cc}{\mathcal{A}}_{can}$ is in 1-1 correspondence with the set $\binom{\zz_1}{\xx_1} \times \binom{\zz_2}{\xx_2}$ and the function $\lambda : (\pi_1(X), \pi_2(X)) \mapsto \pi(X)$ is the bijection between sets $\binom{\zz_1}{\xx_1} \times \binom{\zz_2}{\xx_2}$ and $\binom{\cc}{\mathcal{A}}_{can}$.
\end{fact}

%For a mapping $\pi$ that is an order preserving mapping for $\bb$
%and $\cc$ denote by $\pi(\bb)$ the copy of $\bb$ in $\cc$ induced by mapping $\pi$.

The following Claim states that if $\pi$ is a canonical copy of $\bb$ in $\cc$ and $\tilde{\mathcal{A}}$ is a copy of $\mathcal{A}$ in $\bb$, then  $\pi(\tilde{\mathcal{A}})$  is a canonical copy of $\mathcal{A}$ in $\cc$.

\begin{claim}\label{claim:2}
If $\pi \in \binom{\cc}{\bb}_{can}$ and $\tau \in \binom{\bb}{\aa}$, then $\sigma=\pi \circ \tau \in \binom{\cc}{\aa}_{can}$.
\end{claim}
\begin{proof}
%Let $\bb=(Y, P^{Y}, {L}^{Y}_{1}, {L}^{Y}_{1})$.
Since $\pi: Y \to Z_1 \times Z_2$ is a canonical copy, we have that $\pi=(\pi_1, \pi_2),$ where $\pi_i : Y \to Z_i$ are copies of $Y$ in $Z_{i}$ for $i=1,2$. Define $\sigma_{i}=\pi_{i}\circ \tau$ for $i=1,2$. It is sufficient to prove that for $i=1,2$  $\sigma_{i}$ is order preserving for $\xx$ and $\zz_{i}$.  

Indeed, since $\tau$ preserves $P^{X}, L^{X}_1, L^{X}_2$ and that $\pi_{i}$ preserves $P^{Y}, L^{Y}_i$ for $i=1,2$, we have for any $x,y \in X$ and for $i=1,2$

$$xP^Xy \lr
 \tau(x)P^{Y}\tau(y) \lr  \pi_{i}(\tau(x))P^{Z_{i}}\pi_{i}(\tau(y))\lr \sigma_{i}(x)P^{Z_{i}}\sigma_{i}(y),$$

$$xL^X_{i}y \lr
 \tau(x)L^{Y}_{i}\tau(y) \lr \pi_{i}(\tau(x))L^{Z_{i}}\pi_{i}(\tau(y))\lr \sigma_{i}(x)L^{Z_{i}}\sigma_{i}(y).$$

Consequently, for $i=1,2$, $\sigma_{i}$ is order preserving for $\xx$ and $\zz_{i}$, and $\sigma=(\sigma_1, \sigma_2)$ is a canonical copy of $\aa$ in $\cc$. 
\end{proof}

Our final Claim states that if $\tilde \bb$ is a canonical copy of $\bb$ in $\cc$, and $\tilde{\mathcal{A}}$ is a copy of $\mathcal{A}$ in $\tilde \bb$, then $\tilde{\mathcal{A}}$ is a canonical copy of $\mathcal{A}$ in $\cc$.

\begin{claim}\label{claim:3}
If $\pi \in \binom{\cc}{\bb}_{can}$ and $\sigma \in \binom{\pi(\bb)}{\aa}$, then $\sigma \in \binom{\cc}{\aa}_{can}$.
\end{claim}
\begin{proof}
Since $\pi$ is an isomorphism between $\bb$ and $\pi(\bb)$, the mapping $\pi^{-1}$ exists and is order preserving for $\pi(\bb)$ and $\bb$. Since $\sigma: \aa \to \sigma(\aa) \subseteq \pi(\bb)$ and $\pi^{-1}: \pi(\bb) \to \bb$, the mapping $\tau=\pi^{-1} \circ \sigma: \aa \to \bb$ is well defined. Moreover, $\sigma$ and $\pi^{-1}$ are order preserving, so is also $\tau$. Finally, Claim \ref{claim:2} applied for $\pi$ and $\tau$ gives that $\pi \circ \tau =\sigma$ is a canonical copy of $\mathcal{A}$.
\end{proof}



\section{Proof of Theorem \ref{thm:main}}\label{sec:proof}
Let $\mathcal{A}=(X, P^{X}, L_{1}^{X}, L_{2}^{X})$ and $\bb=(Y,P^{Y}, L_{1}^Y, L_{2}^Y)$ be given. Applying Theorem \ref{thm:product} with $\xx_{i}=(X,P^{X}, L_{i}^X)$ for $i=1,2$ and $\yy_i=(Y,P^{Y}, L_{i}^Y)$ for $i=1,2$ we obtain $\zz_{i}=(Z_{i},P^{Z_{i}}, L_{i}^{Z_{i}})$ for $i=1,2$. 


Set $\cc=\zz_1 \sqcup \zz_2$. Let $\chi: \binom{\cc}{\mathcal{A}} \to \{red,blue\}$ be a colouring.
Since $\binom{\cc}{\mathcal{A}}_{can} \subseteq \binom{\cc}{\mathcal{A}}$, colouring $\chi$ induces $\{red,blue\}$ colouring of $\binom{\cc}{\mathcal{A}}_{can}$. By Fact \ref{fact:lambda},  $\binom{\zz_1}{\xx_1} \times \binom{\zz_2}{\xx_2}$ and  $\binom{\cc}{\mathcal{A}}_{can}$ are in 1-1 correspondence and thus $\chi$ induces a colouring of $\binom{\zz_1}{\xx_1} \times \binom{\zz_2}{\xx_2}$. By a choice of $\zz_1$ and $\zz_2$ (recall that $\zz_i \in PL^{(1)}, \; i=1,2$) there are $\tilde{Y}_i \in \binom{\zz_{i}}{\yy_{i}}$ for $i=1,2$, such that 
$\binom{\tilde{\yy}_1}{\xx_1} \times \binom{\tilde{\yy}_2}{\xx_2}$ is monochromatic and w.l.o.g we assume that all elements of $\binom{\tilde{\yy}_1}{\xx_1} \times \binom{\tilde{\yy}_2}{\xx_2}$ are red.


Let $\pi_{i} : \yy_i \to \zz_i$ be a copy of $\yy_i$ in $\zz_i$, such that $\pi_{i}(\yy_i)=\tilde{\yy_i}$ for $i=1,2$. Then, by Claim \ref{claim:1},
the mapping $\pi : Y \to Z_1 \times Z_2$  defined by 
$\pi(y)=(\pi_1(y), \pi_2(y))$ is a canonical copy of $\bb$ in $\cc$ (see Remark \ref{remark:1}) i.e. $\pi \in \binom{\cc}{\bb}_{can}$. Let $\sigma \in \binom{\pi(\bb)}{\aa}$, then, by Claim \ref{claim:3}, $\sigma \in \binom{\pi(\bb)}{\aa}_{can}$.

Therefore, $\sigma$ is of the form $\sigma(x)=(\sigma_1(x), \sigma_2(x))$, where $\sigma_1 \in \binom{\tilde{\yy_{1}}}{\xx_1}$
and $\sigma_2 \in \binom{\tilde{\yy_{2}}}{\xx_2}$. Since all elements of $\binom{\tilde{\yy_{1}}}{\xx_1} \times \binom{\tilde{\yy_{2}}}{\xx_2}$ are red, we get that the pair $(\sigma_1, \sigma_2)$ and $\sigma$ itself is red. 
Consequently,
every element of $\binom{\pi(\bb)}{\mathcal{A}}$ is colored red.
 
\subsection*{Concluding remarks}
We chose to present the argument for $k=2$ for its notational ease. With the concept of join of two posets replaced with join of $k$ posets, as in definition below,  the proof follows the line of the argument presented in this note.

\begin{definition}
Let $\zz_{i}=(Z_i, P^{Z_i}, L^{Z_i})\in PL^{(1)}$ for $i \in [k]$ and set $C=\Pi_{i=1}^{k}Z_i$.

Define partial order $<_C$ on set $C$  by
$\overline{x}<_{C}\overline{y}$  if $x_{i}P^{Z_{i}}y_{i}$ for all $i \in [k]$.

For all $i \in [k]$ define shifted lexicographic orders $<_{lx_i}$ on set $\Pi_{i=1}^{k}Z_i$, by 

$$\overline{x}<_{lx_i}\overline{y} \lr x_{i+\delta}L^{Z_{i+\delta}}y_{i+\delta},$$
where $\delta$ is the smallest non-negative number $j$, for which $x_{i+j}\neq y_{i+j}$ (with addition mod $k$). Let $\mathcal{L}_{k}^{C}=(<_{lx_1}, <_{lx_2}, \dots, <_{lx_k})$.
Then the  join of $\zz_1, \dots, \zz_k$ is $$\cc=(C, <_C, \mathcal{L}_{k}^C).$$
\end{definition}
 
During preparation of this paper it was brought to our attention that Theorem \ref{thm:main} also follows from the results of Soki\'c~\cite{Sokic}. Alternative proof of Theorem \ref{thm:main} can be deduced from  Theorem 1 in~\cite{Sokic} and follows the same steps as the proof presented in this note.

The original version of this note is available on \href{https://arxiv.org/abs/1608.05290}{arxiv.org}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \bibliographystyle{plain} 
% \bibliography{myBibFile} 
% If you use BibTeX to create a bibliography
% then copy and past the contents of your.bbl file into your.tex file

\begin{thebibliography}{11}

\bibitem{Fouche} W.~L.~Fouch{\'e}.     \newblock Symmetry and the Ramsey degree of posets. \newblock {\em Discrete Math.}, 167/168:309--315, 1997.

\bibitem{Nesetril} J. Ne\v set\v ril. \newblock Ramsey classes and homogeneous structures.
\newblock {\em Combin. Probab. Comput.}, 14(1--2):171--189, 2005.
   
\bibitem{NR} J. Ne\v set\v ril and 
   V. R\"odl.
   \newblock Partitions of finite relational and set systems.
   \newblock {\em J. Combinatorial Theory Ser. A},
   22(3):289--312, 1977.
   
\bibitem{NR2} J. Ne\v set\v ril and 
   V. R\"odl.
   \newblock Ramsey partial orders from acyclic graphs.
   \newblock {\em Order},
   posted on 15 June 2017.
   
\bibitem{PTW} M. Paoli, W. T. Jr. Trotter and J. W. Walker.
\newblock Graphs and orders in Ramsey theory and in dimension theory.
%\newblock {\em Graphs and order},
%      Banff, 1984. 
\newblock {\em NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci.},
147:351--394, Reidel, Dordrecht, 1985.

\bibitem{Promel} H. J. Pr{\"o}mel. 
\newblock {\em Ramsey theory for discrete structures}.
\newblock Springer, Cham, xvi+232, 2013.

\bibitem{SZ} S. Solecki and M. Zhao.
\newblock A Ramsey theorem for partial orders with linear extensions.
\newblock {\em European Journal of Combinatorics}, 60:21--30, 2017.

\bibitem{Sokic} M. Soki\'c.
\newblock Directed graphs and boron trees.
\newblock {\em J. Combin. Theory Ser. A}, 132:142--171, 2015.

\bibitem{Sokic1} M. Soki\'c.
\newblock Ramsey properties of finite posets II.
\newblock {\em Order}, 29(1):31--47, 2012. 

\end{thebibliography}

\end{document}