% EJC papers must begin as follows \documentclass[12pt]{article} \usepackage{e-jc} \specs{P12}{20(1)}{2013} % only use standard LaTeX packages % only include essential packages % 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 % use \sloppy to avoid overly wide text \sloppy % 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} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % if needed include a line break (\\) at an appropriate place in the title \title{ The Ramsey number of loose paths\\ in 3-uniform hypergraphs } % input author, affilliation, address and support information as follows; % the address should include the country, and does not have to include % the street address %\bigskip \author{ Leila Maherani\\ \small Department of Mathematical Sciences\\[-0.8ex] \small Isfahan University of Technology\\[-0.8ex] \small Isfahan, 84156-83111, Iran\\ %\small and\\ %\small School of Mathematics\\ [-0.8ex] %\small Institute for Research in Fundamental Sciences\\[-0.8ex] %\small Tehran, 19395-5746, Iran\\ \footnotesize \texttt{l.maherani@math.iut.ac.ir}\\ \and Golam Reza Omidi$^{\footnotesize\textrm{1}}$\\ \small Department of Mathematical Sciences\\[-0.8ex] \small Isfahan University of Technology\\[-0.8ex] \small Isfahan, 84156-83111, Iran\\ \small and\\ \small School of Mathematics\\ [-0.8ex] \small Institute for Research in Fundamental Sciences \\[-0.8ex] \small Tehran, 19395-5746, Iran\\ \footnotesize \texttt{romidi@cc.iut.ac.ir}\vspace{.1cm}\\ \and Ghaffar Raeisi$^{\footnotesize\textrm{2}}$\\ \small Department of Mathematical Sciences\\[-0.8ex] \small Shahrekord University \\[-0.8ex] \small Shahrekord, P.O.Box 115, Iran\\ \small and\\ \small School of Mathematics\\ [-0.8ex] \small Institute for Research in Fundamental Sciences \\[-0.8ex] \small Tehran, 19395-5746, Iran\\ \footnotesize \texttt{g.raeisi@sci.sku.ac.ir}\\ \and Maryam Shahsiah\\ \small Department of Mathematical Sciences\\[-0.8ex] \small Isfahan University of Technology\\[-0.8ex] \small Isfahan, 84156-83111, Iran\\ %\small and\\ %\small School of Mathematics\\ [-0.8ex] %\small Institute for Research in Fundamental Sciences\\[-0.8ex] %\small Tehran, 19395-5746, Iran\\ \footnotesize \texttt{m.shahsiah@math.iut.ac.ir} } \bigskip \small\date{\dateline{Sep 18, 2012}{Jan 12, 2013}{Jan 21, 2013}\\ \small Mathematics Subject Classifications: 05C15, 05C55, 05C65.} \begin{document} %\includegraphics[width=\textwidth]{pageHeader} \maketitle {\footnotesize\maketitle\footnotetext[1] {This research is partially carried out in the IPM-Isfahan Branch and in part supported by a grant from IPM (No.91050416).}} {\footnotesize\maketitle\footnotetext[2] {This research was in part supported by a grant from IPM (No.91050018).}\vspace{-.7cm}} \begin{abstract} \small Recently, asymptotic values of 2-color Ramsey numbers for loose cycles and also loose paths were determined. Here we determine the 2-color Ramsey number of $3$-uniform loose paths when one of the paths is significantly larger than the other: for every $n\geq \Big\lfloor\frac{5m}{4}\Big\rfloor$, we show that $$R(\mathcal{P}^3_n,\mathcal{P}^3_m)=2n+\Big\lfloor\frac{m+1}{2}\Big\rfloor.$$ % keywords are optional \noindent \textbf{Keywords:} Ramsey Number, Loose Path, Loose Cycle. \phantom{spacer} \end{abstract} %\oint\small \section{\smash{Introduction}\large } A\textit{ hypergraph }$\mathcal{H}$ is a pair $\mathcal{H}=(V,E)$, where $V$ is a finite nonempty set (the set of vertices) and $E$ is a collection of distinct nonempty subsets of $V$ (the set of edges). A \textit{$k$-uniform hypergraph } is a hypergraph such that all its edges have size $k$. For two $k$-uniform hypergraphs $\mathcal{H}$ and $\mathcal{G}$, the \textit{Ramsey number} $R(\mathcal{H},\mathcal{G})$ is the smallest number $N$ such that, in any red-blue coloring of the edges of the complete $k$-uniform hypergraph $K^k_N$ on $N$ vertices there is either a red copy of $\mathcal{H}$ or a blue copy of $\mathcal{G}$. There are several natural definitions for a cycle and a path in a uniform hypergraph. Here we consider the one called loose. A $k$-uniform {\it loose cycle} $\mathcal{C}_n^k$ (shortly, a {\it cycle of length $n$}), is a hypergraph with vertex set $\{v_1,v_2,\ldots,v_{n(k-1)}\}$ and with the set of $n$ edges $e_i=\{v_1,v_2,\ldots, v_k\}+i(k-1)$, $i=0,1,\ldots, n-1$, where we use mod $n(k-1)$ arithmetic and adding a number $t$ to a set $H=\{v_1,v_2,\ldots, v_k\}$ means a shift, i.e. the set obtained by adding $t$ to subscripts of each element of $H$. Similarly, a $k$-uniform {\it loose path} $\mathcal{P}_n^k$ (simply, a {\it path of length $n$}), is a hypergraph with vertex set $\{v_1,v_2,\ldots,v_{n(k-1)+1}\}$ and with the set of $n$ edges $e_i=\{v_1,v_2,\ldots, v_k\}+i(k-1)$, $i=0,1,\ldots, n-1$ and we denote this path by $e_0e_1\cdots e_{n-1}$. For $k=2$ we get the usual definitions of a cycle and a path. In this case, a classical result in graph theory (see \cite{Ramsey number of paths}) states that $R(P_n,P_m)=n+\big\lfloor\frac{m+1}{2}\big\rfloor,$ where $n\geq m\geq 1$. Moreover, the exact values of $R(P_n,C_m)$ and $R(C_n,C_m)$ for positive integers $n$ and $m$ are determined \cite{survey}. For $k=3$ it was proved in \cite{Ramsy number of loose cycle} that $R(\mathcal{C}^3_n, \mathcal{C}^3_n)$, and consequently $R(\mathcal{P}^3_n, \mathcal{P}^3_n)$ and $R(\mathcal{P}^3_n, \mathcal{C}^3_n)$, are asymptotically equal to $\frac{5n}{2}$. Subsequently, Gy\'{a}rf\'{a}s et. al. in \cite{Ramsy number of loose cycle for k-uniform} extended this result to the $k$-uniform loose cycles and proved that $R(\mathcal{C}^k_n, \mathcal{C}^k_n)$, and consequently $R(\mathcal{P}^k_n, \mathcal{P}^k_n)$ and $R(\mathcal{P}^k_n, \mathcal{C}^k_n)$, are asymptotically equal to $\frac{1}{2}(2k-1)n$. For small cases, Gy\'{a}rf\'{a}s et. al. (see \cite{subm}) proved that $R(\mathcal{P}^k_3,\mathcal{P}^k_3)=R(\mathcal{P}^k_3,\mathcal{C}^k_3)=R(\mathcal{C}^k_3,\mathcal{C}^k_3)+1=3k-1$ and $R(\mathcal{P}_4^k,\mathcal{P}_4^k)=R(\mathcal{P}_4^k,\mathcal{C}_4^k)=R(\mathcal{C}_4^k,\mathcal{C}_4^k)+1=4k-2$. To see a survey on Ramsey numbers involving cycles see \cite{Rad}. \medskip It is easy to see that $N=(k-1)n+\lfloor\frac{m+1}{2}\rfloor$ is a lower bound for the Ramsey number $R(\mathcal{P}^k_n,\mathcal{P}^k_m)$. To show this, partition the vertex set of $\mathcal{K}_{N-1}^k$ into parts $A$ and $B$, where $|A|=(k-1)n$ and $|B|=\lfloor\frac{m+1}{2}\rfloor-1$, color all edges that contain a vertex of $B$ blue, and the rest red. Now, this coloring can not contain a red copy of $\mathcal{P}_n^k$, since such a copy has $(k-1)n+1$ vertices. Clearly the longest blue path has length at most $m-1$, which proves our claim. Using the same argument we can see that $N$ and $N-1$ are the lower bounds for $R(\mathcal{P}^k_n,\mathcal{C}^k_m)$ and $R(\mathcal{C}^k_n,\mathcal{C}^k_m)$, respectively. In \cite{subm}, motivated by the above facts and some other results, the authors conjectured that these lower bounds give the exact values of the mentioned Ramsey numbers for $k=3$. In this paper, we consider this problem and we prove that $R(\mathcal{P}^3_n,\mathcal{P}^3_m)=2n+\left\lfloor\frac{m+1}{2}\right\rfloor$ for every $n\geq \lfloor\frac{5m}{4}\rfloor$. Throughout the paper, for a 2-edge coloring of a uniform hypergraph $\mathcal{H}$, say red and blue, we denote by $\mathcal{F}_{red}$ and $\mathcal{F}_{blue}$ the induced hypergraph on edges of colors red and blue, respectively. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\medskip %\vspace{-.1cm} \section{\smash{Preliminaries}\large } In this section, we present some lemmas which are essential in the proof of the main results. \medskip \begin{lemma}\label{No Cn} Let $n\geq m\geq 3$ and $\mathcal{K}^k_{(k-1)n+\lfloor\frac{m+1}{2}\rfloor}$ be 2-edge colored red and blue. If $\mathcal{C}_{n}^k\subseteq \mathcal{F}_{red}$, then either $\mathcal{P}_n^k\subseteq \mathcal{F}_{red}$ or $\mathcal{P}_m^k\subseteq \mathcal{F}_{blue}$. \end{lemma} %%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%% \noindent\textbf{Proof. }Let $e_i=\{v_1,v_2,\ldots, v_k\}+i(k-1)$ (mod $n(k-1)$), $i=0,1,\ldots, n-1$, be the edges of $\mathcal{C}_{n}^k\subseteq \mathcal{F}_{red}$ and $W=\{x_1,x_2,\ldots,x_{\lfloor\frac{m+1}{2}\rfloor}\}$ be the set of the remaining vertices. Set $e_{0}^{\prime}=(e_{0}\setminus\{v_{1}\})\cup\{x_{1}\}$ and for $1\leq i\leq m-1$ let $$ %\begin{eqnarray*} e_i^{\prime}= \left\lbrace \begin{array}{ll} (e_i\setminus \{v_{i(k-1)+1}\})\cup\{x_{\frac{i+1}{2}}\} &\mbox{if~} i~\mbox{is~odd},\vspace{.5 cm}\\ (e_i\setminus\{v_{(i+1)(k-1)+1}\})\cup\{x_{\frac{i+2}{2}}\} &\mbox{if~} i~\mbox{is~even}. \end{array} \right.\vspace{.2 cm} $$ \noindent If one of $e_i^{\prime}$ is red, we have a monochromatic $\mathcal{P}_n^k\subseteq \mathcal{F}_{red}$, otherwise $e_0^{\prime}e_1^{\prime}\ldots e^{\prime}_{m-1}$ form a blue $\mathcal{P}_m^k$, which completes the proof. $\hfill \blacksquare$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \bigskip Let $\mathcal{P}$ be a loose path and $x,y$ be vertices which are not in $\mathcal{P}$. By a {\it $\varpi_{\{v_i,v_j,v_k\}}$-configuration}, we mean a copy of $\mathcal{P}^3_2$ with edges $\{x,v_i,v_j\}$ and \{$v_j,v_k,y\}$ so that $v_l$'s, $l\in\{i,j,k\}$, belong to two consecutive edges of $\mathcal{P}$. The vertices $x$ and $y$ are called the end vertices of this configuration. Using this notation, we have the following lemmas. %\bigskip \begin{lemma}\label{spacial configuration} Let $n\geq 10$, $\mathcal{K}^3_{n}$ be 2-edge colored red and blue and $\mathcal{P}$, say in $\mathcal{F}_{red}$, be a maximum path. Let $A$ be the set of five consecutive vertices of $\mathcal{P}$. % with two consecutive edges $e,e'$. If $W=\{x_1,x_2,x_3\}$ is disjoint from $\mathcal{P}$, then we have a $\varpi_S$-configuration in $\mathcal{F}_{blue}$ with two end vertices in $W$ and $S\subseteq A$. \end{lemma} %%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%% \medskip \noindent\textbf{Proof. }First let $A=e\cup e'$ for two edges $e=\{v_1,v_2,v_3\}$ and $e'=\{v_3,v_4,v_5\}$. Since $\mathcal{P}\subseteq \mathcal{F}_{red}$ is maximal, at least one of the edges $e_1=\{x_1,v_1,v_2\}$ and $e_2=\{v_2,v_3,x_2\}$ must be blue. If both are blue, then $e_1e_2$ is such a configuration. So first let $e_1$ be blue and $e_2$ be red. Maximality of $\mathcal{P}$ implies that at least one of the edges $e_3=\{x_2,v_1,v_4\}$ or $e_4=\{x_3,v_2,v_5\}$ is blue (otherwise, replacing $ee'$ by $e_3e_2e_4$ in $\mathcal{P}$ yields a red path greater than $\mathcal{P}$, a contradiction), and clearly in each case we have a $\varpi_S$-configuration. Now, let $e_1$ be red and $e_2$ be blue. Clearly $e_5=\{v_2,v_4,x_3\}$ is blue and $e_2e_5$ form a $\varpi_S$-configuration. Now let $A=\{v_1,v_2,\ldots,v_5\}$ where $e_1=\{x,v_1,v_2\}$, $e_2=\{v_2,v_3,v_4\}$ and $e_3=\{v_4,v_5,y\}$ are three consecutive edges of $\mathcal{P}$. If $\{x_i,v_2,v_3\}$ is a red edge for some $i\in \{1,2,3\}$, then $\{v_3,v_4,x_j\}$ and $\{v_3,v_5,x_j\}$ are blue for $j\neq i$ and so we are done. By the same argument the theorem is true if $\{x_i,v_3,v_4\}$ is red. Now we may assume $\{v_2,v_3,x_i\}$ and $\{v_3,v_4,x_i\}$ are blue for each $i\in \{1,2,3\}$ and so there is nothing to prove. $\hfill \blacksquare$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ %\bigskip \begin{lemma}\label{P5 or P4} Assume that $n\geq \Big\lfloor\frac{5m}{4}\Big\rfloor$ and $\mathcal{K}^3_{2n+\lfloor\frac{m+1}{2}\rfloor}$ is 2-edge colored red and blue. If $\mathcal{P}\subseteq \mathcal{F}_{blue}$ is a maximum path and $W$, $|W|\geq 5$, is a set of the vertices which are not covered by $\mathcal{P}$, then for every 4 consecutive edges $e_1,e_2,e_3,e_4$ of $\mathcal{P}$ either there is a $\mathcal{P}^3_{5}\subseteq \mathcal{F}_{red}$, say $Q$, between $\lbrace e_1,e_2,e_3,e_4 \rbrace$ and $W$ with end vertices in $W$ and with no the last vertex of $e_4$ as a vertex such that $\vert W\cap V(Q)\vert \leq 5$ or there is a $\mathcal{P}^3_{4}\subseteq \mathcal{F}_{red}$, say $Q$, between $\lbrace e_1,e_2,e_3 \rbrace$ and $W$ with end vertices in $W$ and with no the last vertex of $e_3$ as a vertex such that $\vert W\cap V(Q) \vert \leq 4$. In each of the above cases, each vertex of $W$ except one vertex can be considered as the end vertex of $Q$. \end{lemma} \noindent\textbf{Proof. }Suppose that $e_1$,$e_2$,$e_3$,$e_4$ be four consecutive edges in $\mathcal{P}$. Let $e_i=\{v_{2i-1},v_{2i},v_{2i+1} \}$, $1\leq i\leq 4$, and $W=\{x_1,\dots,x_t\}$ and $T=\{ 1,2,\cdots,t\}$. \bigskip \noindent \textbf{Case 1. }For every $1\leq i,j\leq t$, $\{v_1,v_2,x_i \}$ and $\{v_2,v_3,x_j \}$ are red. \bigskip {\it Subcase 1}. For every $1\leq k,l\leq t$, the edges $\{v_{3},v_{4},x_{k}\}$ and $\{v_{4},v_{5},x_{l}\}$ are red. \medskip For each $\{ i_1, i_2, i_3, i_4 \} \in P_{4}(T)$, edges, $\{x_{i_{1}},v_1,v_2\}$,$\{v_2,x_{i_{2}},v_3\}$,$\{v_3,x_{i_{3}},v_4\}$,$\{v_4,v_5,x_{i_{4}}\} $ make a red $\mathcal{P}^3_{4}$ with end vertices $x_{i_{1}}$ and $x_{i_{4}}$. \medskip {\it Subcase 2}. There exists $1\leq k\leq t$, such that the edge $\{v_{3},v_{4},x_{k}\}$ is blue. \medskip So for each $\{ i_1, i_2, i_3, \} \in P_{3}(T)$ with $k \neq i_2,i_3$, $\{x_{i_{1}} ,v_1,v_2 \}$,$\{v_2,v_3,x_{i_{2}} \}$,$\{x_{i_{2}},v_5,v_4 \}$,\\$\{v_4,v_6,x_{i_{3}}\}$ are the edges of a red desired $\mathcal{P}^3_{4}$ with end vertices $x_{i_{1}}$ and $x_{i_{3}}$. \medskip {\it Subcase 3}. There exists $1\leq k\leq t$, such that the edge $\{v_{4},v_{5},x_{k}\}$ is blue. \medskip If for every $1\leq i,j\leq t$, the edges $\{v_{5},v_{6},x_{i}\}$ and $\{v_{6},v_{7},x_{j}\}$ are red, then for every $\{ i_1, i_2, i_3, i_4 \}\in P_{4}(T)$ with $i_3 \neq k$, we can find a red copy of $\mathcal{P}^3_{5}$ with edges $\{x_{i_{1}},v_1,v_2 \}$,$\{v_2,x_{i_{2}},v_3 \}$,$\{v_3,v_4,x_{i_{3}} \}$,$\{x_{i_{3}},v_5,v_6 \}$,$\{v_6,v_7,x_{i_{4}}\}$ and end vertices $x_{i_{1}}$ and $x_{i_{4}}$. Otherwise there exists $1\leq l\leq t$, such that either $\{v_{5,}v_{6},x_{l}\}$ or $\{v_{6},v_{7},x_{l}\}$ is blue. For the first one, for every $\{ i_1, i_2, i_3, i_4 \}\in P_{4}(T)$ with $i_3 \neq k,l$ and $i_4 \neq l$, $\{x_{i_{1}},v_{1},v_{2}\}$,$\{v_{2},x_{i_{2}},v_{3}\}$,$\{v_{3},v_{4},x_{i_{3}}\}$,$\{x_{i_{3}},v_{7},v_6 \}$,$\{v_6,v_8,x_{i_{4}} \}$ make a red copy of $\mathcal{P}^3_{5}$ with end vertices $x_{i_{1}}$ and $x_{i_{4}}$ and for the second one, for every $\{ i_1, i_2, i_3 \}\in P_{3}(T)$ with $l\neq i_2, i_3$ the edges, $\{x_{i_{1}},v_{1},v_{2},\},\{v_{2},v_{3},x_{i_{2}}\},\{x_{i_{2}},v_{6},v_{5}\},\{v_{5},x_{i_{3}},x_l \}$ make a red $\mathcal{P}^3_{4}$ with end vertices $x_{i_{1}}$ and $y$ where $y\in \{ x_{i_{3}},x_l \}$. \bigskip \noindent \textbf{Case 2. } For some $1\leq i\leq t$, $\{v_{1},v_{2},x_{i}\}$ is blue. \bigskip {\it Subcase 1}. For every $1\leq k,l\leq t$, the edges $\{v_{5},v_{6},x_{k}\}$ and $\{v_{6},v_{7},x_{l}\}$ are red. \medskip For each $\{ i_1, i_2, i_3, i_4 \}\in P_{4}(T)$ with $i_j \neq i$, $1\leq j\leq 4$, the edges, $\{ x_{i_{1}},x_{i},v_{3}\}$, $\{v_{3}, x_{i_{2}},v_{2}\}$, $\{v_{2},v_{4}, x_{i_{3}}\}$, $\{ x_{i_{3}},v_{5},v_6\}$, $\{v_6,v_7, x_{i_{4}} \}$ make a red $\mathcal{P}^3_{5}$ with end vertices $y$, $y\in \{ x_{i_{1}},x_i \}$, and $x_{i_{4}}$. \medskip {\it Subcase 2}. For some $1\leq k\leq t$, $\{v_5,v_6,x_k \}$ is blue. \medskip In this case, for each $\{ i_1, i_2, i_3, i_4 \}\in P_{4}(T)$ with $i_j \neq i$, $1\leq j\leq 4$, and $i_3, i_4 \neq k$, the edges $\{x_{i_{1}},x_{i},v_{3}\}$,$\{v_{3},x_{i_{2}},v_{2}\}$,$\{v_{2},v_{4},x_{i_{3}}\}$,$\{x_{i_{3}},v_{7},v_6 \}$, $\{v_6,v_8,x_{i_{4}}\}$ make a red $\mathcal{P}^3_{5}$ with end vertices $y$, $y\in \{ x_{i_{1}},x_i \}$, and $x_{i_{4}}$. \medskip {\it Subcase 3}. For some $1\leq k\leq t$, $\{v_6,v_7,x_k \}$ is blue. \medskip In this case, for each $\{ i_1, i_2, i_3 \}\in P_{3}(T)$ with $i_j \neq i$, $1\leq j\leq 3$, and $i_2, i_3 \neq k$, the edges $\{x_{i_{1}},x_{i},v_{3}\}$,$\{v_3,v_{2},x_{i_{2}}\}$,$\{x_{i_{2}},v_{4},v_{6}\}$,$\{v_6,v_5,x_{i_{3}}\}$ make a red $\mathcal{P}^3_{4}$ with end vertices $y$, $y\in \{ x_{i_{1}},x_i \}$, and $x_{i_3}$. %%%%%%#######################################################################################3 \bigskip \noindent \textbf{Case 3. } For some $1\leq i\leq t$, $\{v_{2},v_{3,}x_{i}\}$ is blue. \medskip {\it Subcase 1}. For every $1\leq k,l\leq t$, the edges $\{v_{3},v_{4},x_{k}\}$ and $\{v_{4},v_{5},x_{l}\}$ are red. \medskip For each $\{ i_1, i_2, i_3 \}\in P_{3}(T)$ with $i_j \neq i$, $1\leq j\leq 3$, $\{x_{i_{1}},x_i,v_1 \}$,$\{v_1,v_2,x_{i_{2}} \}$, $\{x_{i_{2}},v_3,v_4 \}$,$\{v_4,v_5,x_{i_{3}} \}$ are the edges of a red $\mathcal{P}^3_{4}$ with end vertices $y$, $y\in \{ x_{i_{1}},x_i \}$, and $x_{i_{3}}$. \medskip {\it Subcase 2}. For some $1\leq k\leq t$, $\{v_3,v_4,x_k \}$ is blue. \medskip In this case, for each $\{ i_1, i_2, i_3 \}\in P_{3}(T)$ with $i_j \neq i$, $1\leq j\leq 3$, and $i_2, i_3 \neq k$, the edges, $\{x_{i_{1}},x_i,v_1 \}$,$\{v_1,v_2,x_{i_{2}} \}$,$\{x_{i_{2}},v_5,v_4 \}$,$\{v_4,v_6,x_{i_{3}} \}$ make a red copy of $\mathcal{P}^3_{4}$ with end vertices $y$, $y\in \{ x_{i_{1}},x_i \}$, and $x_{i_3}$. \medskip {\it Subcase 3}. For some $1\leq k\leq t$, $\{v_4,v_5,x_k \}$ is blue. \medskip If for every $1\leq l,h\leq t$, the edges $\{v_{5},v_{6},x_{l}\}$ and $\{v_{6},v_{7},x_{h}\}$ are red, then for each $\{ i_1, i_2, i_3, i_4 \}\in P_{4}(T)$ with $i_j \neq i$, $1\leq j\leq 4$, and $i_3\neq k$, the edges, $\{x_{i_{1}},x_i,v_1 \}$,$\{v_1,x_{i_{2}},v_2 \}$,$\{v_2,v_4,x_{i_{3}} \}$,$\{x_{i_{3}},v_5,v_6 \},\{v_6,v_7,x_{i_{4}} \}$ make a red $\mathcal{P}^3_{5}$ with end vertices $y$, $y\in \{ x_{i_{1}},x_i \}$, and $x_{i_{4}}$. Otherwise there exists $1\leq l\leq t$, such that either $\{v_{5,}v_{6},x_{l}\}$ or $\{v_{6},v_{7},x_{l}\}$ is blue. For the first one, for each $\{ i_1, i_2, i_3, i_4 \}\in P_{4}(T)$ with $i_j \neq i$, $1\leq j\leq 4$, $i_3\neq k,l$ and $i_4\neq l$, the edges $\{x_{i_{1}},x_{i},v_{1}\}$,$\{v_{1},x_{i_{2}},v_{2}\}$,$\{v_{2},v_{4},x_{i_{3}}\}$, $\{x_{i_{3}},v_{7},v_6 \}$,$\{v_6,v_8,x_{i_{4}}\}$ make a red copy of $\mathcal{P}^3_{5}$ with end vertices $y$, $y\in \{ x_{i_{1}},x_i \}$ and $x_{i_4}$. For the second one, for every $\{ i_1, i_2, i_3 \}\in P_{3}(T)$ with $i_j \neq i$, $1\leq j\leq 3$, and $i_2, i_3 \neq l$, $\{\{x_{i_{1}},x_{i},v_{1}\},\{v_{1},v_{2},x_{i_{2}}\},\{x_{i_{2}},v_{4},v_{6}\}, \{v_{6},v_{5},x_{i_{3}} \}\}$ is the set of the edges of a red $\mathcal{P}^3_{4}$ with end vertices $y$, $y\in \{ x_{i_{1}},x_i \}$, and $x_{i_3}$. These observations complete the proof. $\hfill \blacksquare$ %\bigskip \section{Main Results} In this section, we prove that $R(\mathcal{P}^3_n,\mathcal{P}^3_m)=2n+\left\lfloor\frac{m+1}{2}\right\rfloor$ for every $n\geq\lfloor\frac{5m}{4}\rfloor$. First we present several lemmas which will be our main tools in establishing the main theorem. %\bigskip \begin{lemma}\label{pm-1 implies pn-1} Assume that $n= \left\lfloor\frac{5m}{4}\right\rfloor$ and $\mathcal{K}^3_{2n+\lfloor\frac{m+1}{2}\rfloor}$ is 2-edge colored red and blue. If $\mathcal{P}=\mathcal{P}^3_{m-1}$ is a maximum blue path, then $\mathcal{P}^3_{n-1}\subseteq \mathcal{F}_{red}$. \end{lemma} %%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%% \noindent\textbf{Proof. }Let $t=2n+\lfloor\frac{m+1}{2}\rfloor$ and $\mathcal{P}=e_1e_2\ldots e_{m-1}$ be a copy of $\mathcal{P}_{m-1}^3\subseteq \mathcal{F}_{blue}$ with edges $e_i=\{v_1,v_2,v_3\}+2(i-1)$, $i=1,\ldots, m-1$. Set $W=V(\mathcal{K}^3_t)\setminus V(\mathcal{P})$. Using Lemma \ref{P5 or P4} there is a red path $Q_1$ with end vertices $x_1$ and $y_1$ in $W_1=W$ between $E'_1$ and $W_1$ where $E_1=\{e_i: i_1=1\leq i \leq 4\}$, $\bar{E}_1=E_1\setminus \{e_4\}$ and $E'_1\in \{E_1,\bar{E}_1\}$. Set $i_2=\min\{j: j\in\{i_1+3,i_1+4\}, e_j\not\in E'_1\}$, $E_2=\{e_i: i_2\leq i\leq i_2+3\}$ and $\bar{E}_2=E_2\setminus \{e_{i_2+3}\}$ and $W_2=(W\setminus V(Q))\cup \{x_1,y_1\}$. Again using Lemma \ref{P5 or P4} there is a red path $Q_2$ between $E'_2$ and $W_2$ such that $Q_1\cup Q_2$ is a red path with end vertices $x_2, y_2$ in $W_2$ where $E'_2\in \{E_2, \bar{E}_2\}$ and again set $i_3=\min\{j: j\in\{i_2+3,i_2+4\}, e_j\not\in E'_2\}$, $E_3=\{e_i: i_3\leq i\leq i_3+3\}$, $\bar{E}_3=E_3\setminus \{e_{i_3+3}\}$ and $W_3=(W\setminus V(Q_1\cup Q_2))\cup \{x_2,y_2\}$. Since $|W|\geq m$, using Lemma \ref{P5 or P4} by continuing the above process we can partition $E(\mathcal{P})\setminus\{e_{m-1}\}$ into classes $E'_i$th, $|E'_i|\in\{3,4\}$ and at most one class of size $r\leq 3$ of the last edges such that for each $i$, there is a red $Q_i=\mathcal{P}^3_5$ (resp. $Q_i=\mathcal{P}^3_4$) between $E'_i$ and $W$ with the properties in Lemma \ref{P5 or P4} if $|E'_i|=4$ (resp. $|E'_i|=3$) and $\mathcal{P}'=\Large\cup Q_i$ is a red path with end vertices $x, y$ in $W$. Let $l_1=|\{i: \mid E'_i\mid=4\}|$ and $l_2=|\{i: \mid E'_i\mid=3\}|$. So $m-2=4l_1+3l_2+r$, $0\leq r\leq 3$ and $\mathcal{P}^{\prime}$ has $5l_1+4l_2$ edges. One can easily check that $5l_1+4l_2\geq \frac{5}{4}(m-2-r)$. Also we have $$|W\cap V(\mathcal{P}^{\prime})|\leq 4l_1+3l_2+1=m-1-r.$$ Let $T=V(\mathcal{K}^3_t)\setminus (V(\mathcal{P})\cup V(\mathcal{P}'))$ and suppose that $m=4k+p$ for some $p$, $0\leq p\leq 4$. Therefore $|T|\geq r+2$ if $p=0,1$ and $|T|\geq r+1$ if $p=2,3$. Now we consider the following cases. \bigskip \noindent \textbf{Case 1. }$r=0$. \medskip Clearly $|T|\geq 1$ and it is easy to see that $\mathcal{P}^{\prime}$ contains at least $n-2$ edges. Let $\{u\}\subseteq T$. The maximality of $\mathcal{P}$ implies that the edge $e=\{v_{2m-1},x,u\}$ is red and hence $\mathcal{P}^{\prime}\cup \{e\}$ is a red copy of $\mathcal{P}^3_{n-1}$. \bigskip \noindent \textbf{Case 2. }$r=1$. \medskip In this case, $|T|\geq 2$ and it is easy to see that $\mathcal{P}^{\prime}$ contains at least $n-3$ edges. Let $\{u,v\}\subseteq T$. Clearly $\mathcal{P}^{\prime}\cup \{\{v_{2m-2},x,u\}, \{v_{2m-1},u,v\}\}$ is a red copy of $\mathcal{P}^3_{n-1}$. \bigskip \noindent \textbf{Case 3. }$r=2$. \medskip It is easily seen that $|T|\geq 3$ and $\mathcal{P}^{\prime}$ contains at least $n-5$ edges. Let $T'=\{u,v,w\}\subseteq T$. Since $V(\mathcal{P}^{\prime})\cap V(e_{m-3}\cup e_{m-2})=\emptyset$ by Lemma \ref{spacial configuration} there is a red $\varpi_S$-configuration with $S\subset e_{m-3}\cup e_{m-2}$ and its end vertices in $T'$, say $u$ and $v$. The maximality of $\mathcal{P}$ implies that the edges $\{v_{2m-2},x,u\}$ and $\{v_{2m-1},v,w\}$ are red and clearly we have a red $\mathcal{P}^3_{n-1}$. \bigskip \noindent \textbf{Case 4. }$r=3$. \medskip In this case, for $p\in \{2,3\}$ we have $|T|\geq 4$ and $\mathcal{P}^{\prime}$ contains at least $n-5$ edges. Using an argument similar to case 3 we can complete the proof. Now let $p\in \{0,1\}$. Then $|T|\geq 5$ and $\mathcal{P}^{\prime}$ contains at least $n-6$ edges. Set $T'=\{u,v,w,z,t\}\subseteq T$. By Lemma \ref{spacial configuration}, there is a $\varpi_S$-configuration $C$ with $S\subseteq V(e_{m-3}\cup e_{m-2})$ and end vertices in $T'$, say $u$ and $v$. Clearly $\mathcal{P}^{\prime}\cup\{\{y,w,v_{2m-2}\},\{v_{2m-2},z,t\},\{v_{2m-1},t,u\}\}\cup C$ is a red $\mathcal{P}^3_{n-1}$. These observations complete the proof. $\hfill \blacksquare$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\bigskip \begin{lemma}\label{a} Let $n\geq \Big\lfloor\frac{5m}{4}\Big\rfloor$ and $\mathcal{K}^3_{2n+\lfloor\frac{m+1}{2}\rfloor}$ be 2-edge colored red and blue. If $\mathcal{P}^3_{n-1}\subseteq \mathcal{F}_{red}$ be a maximum path, then $\mathcal{P}^3_{m}\subseteq \mathcal{F}_{blue}$. \end{lemma} %%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%% \noindent\textbf{Proof. }Let $t=2n+\lfloor\frac{m+1}{2}\rfloor$ and $\mathcal{P}=e_1e_2\ldots e_{n-1}$ be a copy of $\mathcal{P}_{n-1}^3\subseteq \mathcal{F}_{red}$ with end edges $e_1=\{v_1,v_2,v_3\}$ and $e_{n-1}=\{v_{2n-3},v_{2n-2},v_{2n-1}\}$. By Lemma \ref{No Cn}, we may assume that the subhypergraph induced by $V(\mathcal{P})$ does not have a red copy of $\mathcal{C}^3_n$. Let $W=V(\mathcal{K}^3_t)\setminus V(\mathcal{P})$ and let $2n-2=5q+h$ where $0\leq h<5$. Partition the set $V(\mathcal{P})\setminus \{v_1\}$ into $q$ classes $A_1,A_2,\ldots,A_q$ of size five and one class $A_{q+1}=\{v_{2n-h},\ldots,v_{2n-2},v_{2n-1}\}$ of size $h$ if $h>0$, so that each class contains consecutive vertices of $\mathcal{P}$. Using Lemma \ref{spacial configuration}, there is a blue $\varpi_{S_1}$-configuration, $\bar{c}_1$, with the set of end vertices $E_1\subseteq W$ and $S_1\subseteq A_1$. Let $x_1\in E_1$ and $B_1$ be a 2-subset of $W\setminus E_1$. Again by Lemma \ref{spacial configuration}, there is a blue $\varpi_{S_2}$-configuration, $\bar{c}_2$, with the set of end vertices $E_2\subseteq (B_1\cup \{x_1\})$ and $S_2\subseteq A_2$. If $x_1\not\in E_2$, then let $\bar{c}_3$ be a blue $\varpi_{S_3}$-configuration with the set of end vertices $E_3\subseteq\{x_1,y,z\}$ and $S_3\subseteq A_3$ where $y\in B_1$ and $z\in W\setminus (E_1\cup E_2)$. If $x_1\in E_2$, then let $\bar{c}_3$ be a blue $\varpi_{S_3}$-configuration with the set of end vertices $E_3\subseteq\{x_2,y,z\}$ and $S_3\subseteq A_3$ where $x_2\in E_2\setminus\{x_1\}$ and $\{y,z\}\subseteq W\setminus(E_1\cup E_2)$. We continue this process to find the set of $\{\bar{c}_1,\bar{c}_2,\ldots,\bar{c}_{q'}\}$ of configurations. When this process terminate, %by combining these configurations we have the paths $\mathcal{P}_{l''}$ and $\mathcal{P}_{l'}$ where $l''\geq l'\geq 0$ and $l''+l'=2q'$. Let $x'',y''$ (resp. $x',y'$ if $l'> 0$) be the end vertices of $\mathcal{P}_{l''}$ (resp. $\mathcal{P}_{l'}$) in $W$. Let $T=V(\mathcal{K}^3_t)\setminus (V(\mathcal{P})\cup V(\mathcal{P}_{l''})\cup V(\mathcal{P}_{l'}))$. Clearly $|T|=\lfloor\frac{m+1}{2}\rfloor+1-(q'+i)$ where $i=1$ if $l'=0$ and $i=2$ if $l'>0$. Assume $m=4k+r$ for some $r$, $0\leq r\leq 3$. We have the following cases. \bigskip \noindent \textbf{Case 1. }$r=0$. \medskip Since $q\geq 2k-1$, we have $2q'\geq m-2$. On the other hand, $|W|=\lfloor\frac{m+1}{2}\rfloor+1$ and so $2q'\leq m$. If $2q'=m$, then $l'=0$ and so $\mathcal{P}_{l''=m}$ is a blue path. Now we may assume that $2q'=m-2$, and one can easily check that the vertices $\{v_{2n-3},v_{2n-2},v_{2n-1}\}$ are not used in $\mathcal{P}_{l''}\cup\mathcal{P}_{l'}$. First let $l'=0$. Then $|T|=1$ and we may assume $T=\{u\}$. Now using the maximality of $\mathcal{P}$ and the fact that $\mathcal{C}^3_{n}\nsubseteq \mathcal{F}_{red}$, $\mathcal{P}_{l''}\cup\{\{v_{2n-2},y'',u\}, \{v_{2n-1},u,v_1\}\}$ is a blue $\mathcal{P}^3_{m}$. For $l'> 0$, $\mathcal{P}_{l''}\cup\{\{v_{2n-2},y'',x'\}\}\cup\mathcal{P}_{l'}\cup\{\{v_{2n-1},y',v_1\}\}$ is a blue $\mathcal{P}^3_{m}$. \bigskip \noindent \textbf{Case 2. }$r=1$. \medskip Since $\mid W\mid=\lfloor \frac{m+1}{2}\rfloor+1$, $2q'\leq m+1$ and if the equality holds, then $l'=0$. On the other hand, $q\geq 2k$ and so $2q'\geq m-1$. Hence $2q'\in \{m+1,m-1\}$. If $2q'=m+1$, then $l'=0$ and there is a blue $\mathcal{P}^3_{m+1}$. Now let $2q'=m-1$. If $l'=0$, then $|T|=1$, so $T=\{u\}$ and hence $\mathcal{P}_{l''}\cup\{\{v_1,u,y''\}\}$ is a blue $\mathcal{P}^3_{m}$. If $l'>0$, then $\mathcal{P}_{l''}\cup \{\{v_1,y'',x'\}\}\cup \mathcal{P}_{l'}$ is a blue $\mathcal{P}^3_{m}$. \bigskip \noindent \textbf{Case 3. }$r=2$. \medskip Using an argument similar to the case 1, we have $2q'\in \{m,m-2\}$ and if $2q'=m$, then $l'=0$ and we have a blue $\mathcal{P}_{l''=m}$. Again by an argument similar to the case 1 we have a blue $\mathcal{P}^3_m$. \bigskip \noindent \textbf{Case 4. }$r=3$. \medskip In this case, partition $V(\mathcal{P})\setminus \{v_1,v_2\}$ into $\lfloor\frac{2n-3}{5}\rfloor$ classes of size five and possibly one class of size at most four. Then we repeat the mentioned process in the first of the proof to find blue paths $\mathcal{P}_{l''}$ and $\mathcal{P}_{l'}$ with $l''\geq l'\geq 0$ and $l''+l'=2q'$. Again using a similar argument in case 1, we have $2q'\in \{m+1,m-1,m-3\}$. If $2q'=m+1$, then we have $l'=0$ and so there is a blue $\mathcal{P}^3_{m+1}$. For $2q'=m-1$, the assertion holds by an argument similar to the case 2. Now let $2q'=m-3$. If $l'=0$, then $|T|=2$, so $T=\{u,v\}$ and hence $\mathcal{P}_{l''}\cup \{\{v_{2n-2},v_2,y''\}, \{v_{2n-2},v,u\}, \{u,v_1,v_{2n-1}\}\}$ is a blue $\mathcal{P}^3_{m}$ (note that $\{v_{2n-3},v_{2n-2},v_{2n-1}\}\cap V(\mathcal{P}_{l''})=\emptyset $). If $l'>0$, then $|T|=1$, so $T=\{u\}$ and hence $\mathcal{P}_{l''}\cup \{\{v_{2n-2},v_2,y''\}, \{v_{2n-2},x',u\}\}\cup \mathcal{P}_{l'}\cup \{\{y',v_1,v_{2n-1}\}\}$ is a blue $\mathcal{P}^3_{m}$ and the proof is completed. $\hfill\blacksquare$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\bigskip \begin{theorem}\label{main theorem} For every $n\geq \left\lfloor\frac{5m}{4}\right\rfloor$, $$R(\mathcal{P}^3_n,\mathcal{P}^3_m)=2n+\Big\lfloor\frac{m+1}{2}\Big\rfloor.$$ \end{theorem} %%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%% \noindent\textbf{Proof. }We prove the theorem by induction on $m+n$. The proof of the case $m=n=1$ is trivial. Suppose that for $m^{\prime}+n^{\prime}< m+n$ with $n'\geq \lfloor\frac{5m'}{4}\rfloor$, $R(\mathcal{P}^3_{n^{\prime}}, \mathcal{P}^3_{m^{\prime}})=2n^{\prime}+ \left\lfloor\frac{m^{\prime}+1}{2}\right\rfloor.$ Now, let $n\geq \left\lfloor\frac{5m}{4}\right\rfloor$ and let $\mathcal{K}^3_{2n+\lfloor\frac{m+1}{2}\rfloor}$ be 2-edge colored red and blue. We may assume there is no red copy of $\mathcal{P}^3_n$ and no blue copy of $\mathcal{P}^3_m$. Consider the following cases. \bigskip \noindent \textbf{Case 1. } $n= \left\lfloor\frac{5m}{4}\right\rfloor$. \medskip Since $R(\mathcal{P}^3_{n-1},\mathcal{P}^3_{m-1})= 2(n-1)+\left\lfloor\frac{m}{2}\right\rfloor< 2n+\left\lfloor\frac{m+1}{2}\right\rfloor$ by induction hypothesis, then either there is a $\mathcal{P}^3_{n-1}\subseteq \mathcal{F}_{red}$ or a $\mathcal{P}^3_{m-1}\subseteq \mathcal{F}_{blue}$. If we have a red copy of $\mathcal{P}^3_{n-1}$, then by Lemma \ref {a} we have a $\mathcal{P}^3_{m}\subseteq \mathcal{F}_{blue}$. Now assume that there is a blue copy of $\mathcal{P}^3_{m-1}$. %then Since $n-1= \Big\lfloor\frac{5(m-1)}{4}\Big\rfloor$, Lemma \ref{pm-1 implies pn-1} implies that $\mathcal{P}^3_{n-1}\subseteq \mathcal{F}_{red}$ and using Lemma \ref {a} we have $\mathcal{P}^3_{m}\subseteq \mathcal{F}_{blue}$, a contradiction. \bigskip \noindent \textbf{Case 2. }$n> \left\lfloor\frac{5m}{4}\right\rfloor$. \medskip In this case, $n-1\geq \left\lfloor\frac{5m}{4}\right\rfloor$ and since $R(\mathcal{P}^3_{n-1},\mathcal{P}^3_{m})= 2(n-1)+\left\lfloor\frac{m+1}{2}\right\rfloor< 2n+\left\lfloor\frac{m+1}{2}\right\rfloor$, by induction hypothesis we have a $\mathcal{P}^3_{n-1}\subseteq \mathcal{F}_{red}$. Using Lemma \ref {a} we have a $\mathcal{P}^3_{m}\subseteq \mathcal{F}_{blue}$ and it completes the proof. $\hfill\blacksquare$ %\bigskip \subsection*{Acknowledgments} \emph{\emph{The authors appreciate the discussions on the subject of this paper with A. Gy\'arf\'as. Also the third author would like to thanks Shahrekord University for its support. } } %\bigskip %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\footnotesize \begin{thebibliography}{10} %\vspace{.6 cm} \bibitem{Ramsey number of paths} L. Gerencs\'{e}r, A. Gy\'{a}rf\'{a}s, On Ramsey-type problems, {\it Ann. Univ. Sci. Budapest, E\"{o}tv\"{o}s Sect. Math.} {\bf 10} (1967), 167-170. \bibitem{subm} A. Gy\'{a}rf\'{a}s, G. Raeisi, The Ramsey number of loose triangles and quadrangles in hypergraphs, {\it Electron. J. Combin.} {\bf 19} (2012), no. 2, \#R30. \bibitem{Ramsy number of loose cycle for k-uniform} A. 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