% 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. 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