\documentclass[12pt]{article} \usepackage{e-jc} \specs{P3.11}{22(3)}{2015} \usepackage{graphicx} \usepackage{enumerate} \usepackage[vlined,ruled]{algorithm2e} \usepackage{amsmath} \usepackage{tikz} \usepackage{float} \usepackage[colorlinks=true,citecolor=black,linkcolor=black,urlcolor=blue]{hyperref} \input amssym.def \input amssym.tex \bibliographystyle{plain} %\renewcommand{\arraystretch}{1.1} \newcommand{\howmany}{$28$ } %%%%%%% %\setlength{\textwidth}{6.3in} %\setlength{\textheight}{8.7in} %\setlength{\topmargin}{0pt} %\setlength{\headsep}{0pt} %\setlength{\headheight}{0pt} %\setlength{\oddsidemargin}{0pt} %\setlength{\evensidemargin}{0pt} %%%%%%% \begin{filecontents}{log.data} 3 2.585 4 4.170 5 5.426 6 6.672 7 7.679 8 8.140 9 9.142 10 9.640 \end{filecontents} \begin{document} \title{\bf New Lower Bounds for 28 Classical Ramsey Numbers} \author{ Geoffrey Exoo \\ \small Department of Mathematics and Computer Science \\[-0.8ex] \small Indiana State University \\[-0.8ex] \small Terre Haute, IN 47809, U.S.A. \\ \small\tt ge@cs.indstate.edu \\ \\ Milos Tatarevic \\ \small Alameda, CA 94501, U.S.A \\ \small\tt milos.tatarevic@gmail.com \\ } \date{\dateline{May 12, 2015}{Jun 24, 2015}{Jul 17, 2015}\\ \small Mathematics Subject Classifications: 05C55, 05D10} \maketitle \begin{abstract} We establish new lower bounds for \howmany classical two and three color Ramsey numbers, and describe the heuristic search procedures used. Several of the new three color bounds are derived from the two color constructions; specifically, we were able to use $(5,k)$-colorings to obtain new $(3,3,k)$-colorings, and $(7,k)$-colorings to obtain new $(3,4,k)$-colorings. Some of the other new constructions in the paper are derived from two well known colorings: the Paley coloring of $K_{101}$ and the cubic coloring of $K_{127}$. \end{abstract} \section{Introduction} The classical Ramsey number $R(k_{1},k_{2},\dots,k_{r})$ is the smallest integer $n$ such that in any $r$-coloring of the edges of the complete graph $K_n$ there is a monochromatic copy of $K_{k_{c}}$ for some $1\leq c \leq r$. In this note, we examine colorings for $r=2$ and $r=3$, describe some new variations on computer construction methods and are able to improve the lower bounds for \howmany classical Ramsey numbers. When discussing the procedures for improving lower bounds, we use the term {\em bad subgraphs} to mean monochromatic complete graphs of order $k_{c}$ in any of the colors $1\leq c \leq r$. The reader is referred to Radziszowski's survey on Small Ramsey Numbers \cite{survey} for basic terminology related to the problem. The survey contains a comprehensive summary of the current state of the art. In this report the emphasis is on establishing new lower bounds for a range of Ramsey numbers of the form $R(k_1,k_2)$. Our primary focus is on cases where $4 \leq k_1 \leq 6$ and where current lower bounds are between $100$ and $200$, though several bounds outside this range are also presented. Some of these colorings are then used to obtain lower bounds for three-color classical Ramsey numbers. The new values for two- and three-color numbers are listed in Tables 1 and 2, respectively. The graphs were constructed by three variations on a basic method. These three variations involve searching through colorings such that the adjacency (coloring) matrix is partitioned into (1) cyclic orbits, (2) square sub-blocks with cyclic orbits, or (3) single edge orbits. In the first case we get circle colorings, which have been used many times in this context \cite{survey}. In the second case we have colorings such that the adjacency matrix can be partitioned into square circulant submatrices (of the same size); and in the third case we have arbitrary coloring matrices. Given a particular Ramsey number, $R(k_1,k_2)$ and a value of $n$, the goal is to find a good $(k_1,k_2)$-coloring of $K_n$. We begin by searching for circle colorings. If that search succeeds, then we might increase $n$, and repeat. If it doesn't succeed and if $n$ is not prime, then we expand our search space by considering all integer factorizations $n = m d$, and search for circulant block colorings where the adjacency matrix is partitioned into an $m \times m$ array of $d \times d$ circulant blocks. If neither of these searches succeeds, but if either of them produces colorings with a relatively small number of bad subgraphs, then the coloring is used as a starting configuration and a search which recolors single edges is applied. The decision as to whether a given number of bad subgraphs is small enough to merit further investigation is made empirically. In a further variation on this method, a few of the new colorings presented here were obtained by using one of two well known circulant colorings to create initial colorings, and then applying the single edge recoloring procedure. These two special colorings are the Paley (or quadratic) coloring of $K_{101}$ and the cubic coloring of $K_{127}$. Recall that the former coloring was used to establish the current lower bound for $R(6,6)$ \cite{kalb}, while the latter was used to establish lower bounds for both $R(4,12)$ \cite{sll} and $R(4,4,4)$ \cite{hillirving}. The cubic residue graph of order $127$ may be of further interest, since it was conjectured by the first author to be a Folkman graph i.e., a $K_4$-free graph such that in any two-coloring of the edges, there is a monochromatic $K_3$ (see, for example, Conjecture 4.4 in \cite{xurad} which contains a fascinating discussion of this and related problems). \section{New Lower Bounds} As indicated above, we present two tables, one each for two-color and three-color lower bounds. A few of the cases listed were obtained using {\it ad hoc} methods, which are outlined below. For the other cases, we stayed fairly close to the general approach described above. In the column labeled {\it method} we indicate how we obtained the initial coloring that gave, or led to, the indicated lower bound. Since these were computer searches, and the success of such searches depends to some extent on the computer time available, we attempted to assign approximately the same time to each problem. \begin{table}[ht] \centering \begin{tabular}{lccc} Ramsey Number & Old Bound & New Bound & Method \\ \hline R(4, 8) & 58 & 59 & circulant blocks \\ R(4,11) & 98 & 102 & circulant blocks \\ R(4,13) & 133 & 138 & cubic(127) \\ R(4,14) & 141 & 147 & cubic(127) \\ R(4,15) & 153 & 155 & cubic(127) \\ R(4,16) & 164 & 166 & cubic(127) \\ \hline R(5,10) & 144 & 149 & circulant-minus \\ R(5,11) & 171 & 174 & circulant \\ R(5,12) & 191 & 194 & circulant \\ R(5,13) & 213 & 218 & circulant \\ R(5,14) & 239 & 242 & circulant \\ R(5,15) & 265 & 269 & circulant \\ R(5,16) & 290 & 293 & circulant \\ \hline R(6, 7) & 113 & 115 & Paley(101) \\ R(6, 8) & 132 & 134 & Paley(101) \\ R(6, 9) & 169 & 175 & circulant \\ R(6,10) & 179 & 185 & circulant \\ \hline R(7, 9) & 241 & 242 & circulant \\ \hline \end{tabular} \caption{New lower bounds for 2-color classical Ramsey numbers.} \end{table} In Table 2, new lower bounds for $3$-color Ramsey numbers are shown. Some of these colorings were found by using one of our $(5,k)$-colorings, and {\em splitting} the $K_5$-free color graph into two $K_3$-free color graphs, therby obtaining a $(3,3,k)$-coloring. In two other cases we used one of our $(7,k)$-colorings to obtain a $(3,4,k)$-coloring. Here we split the $K_7$-free color graph into a $K_3$-free graph and a $K_4$-free graph. \begin{table}[ht] \centering \begin{tabular}{lccc} Ramsey Number & Old Bound & New Bound & Method \\ \hline R(3,3,6) & 60 & 61 & circulant blocks \\ R(3,3,10) & 147 & 150 & circulant \\ R(3,3,11) & 162 & 174 & splitting \\ R(3,3,12) & 185 & 194 & splitting \\ R(3,3,13) & 212 & 217 & splitting \\ R(3,3,14) & 233 & 242 & splitting \\ R(3,3,15) & & 269 & splitting \\ R(3,3,16) & & 291 & splitting \\ \hline R(3,4,7) & 145 & 152 & splitting \\ R(3,4,9) & 229 & 242 & splitting \\ \hline \end{tabular} \caption{New lower bounds for 3-color classical Ramsey numbers.} \end{table} All of the colorings described in this paper are available at \begin{center} \url{www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p11/data} \end{center} and \begin{center} \url{http://cs.indstate.edu/ge/RAMSEY/ExTa}. \end{center} \section{Search Methods} Our basic method begins by considering colorings where the adjacency matrix is either a circulant or can be partitioned into square circulant submatrices of the same size. For a graph of order $n$, let $m$ be a positive integer such that $m\mid n$. We can partition the coloring matrix $A$ into $m \times m$ square blocks, as follows \begin{equation} \label{block} A=\begin{bmatrix} %\left[\begin{array}{cccc} C_{1} & C_{2} & \cdots & C_{m}\\ C_{2}^{t} & C_{m+1} & \cdots & C_{2m-1}\\ \vdots & \vdots & \ddots & \vdots\\ C_{m}^{t} & C_{2m-1}^{t} & \cdots & C_{\binom{m+1}{2}} \end{bmatrix}, %\end{array}\right], \end{equation} \noindent and constrain each of the blocks $C_{i}$ to be a $d\times d$ circulant matrix, with $d=n/m$. Note that the diagonal blocks must be symmetric circulant matrices. For $m=1$, the entire coloring matrix of $K_{n}$ is a symmetric circulant. For $m>1$, the off-diagonal matrices can be either symmetric or asymmetric circulant blocks. The former case significantly reduces the size of the search space, which makes things faster, but may eliminate some good colorings. When off-diagonal blocks are asymmetric, the number of required color choices is \begin{equation} \label{block-sym} m \lfloor d/2 \rfloor + \binom{m}{2} d. \end{equation} Whereas, if they are required to be symmetric, the number of color choices is \begin{equation} \label{block-asym} m \lfloor d/2 \rfloor + \binom{m}{2} (\lfloor d/2 \rfloor + 1). \end{equation} Let $V$ denote the vector of color assignments that determines a coloring of $K_{n}$ where the adjacency matrix is partitioned into circulant blocks, as in (\ref{block}). When $m=1$ (and we have a circle coloring), the coloring can be completely specified by a vector $V=(a_{1},\dots,a_{\lfloor n/2 \rfloor})$. For $m>1$, the length of the coloring vector $V$ is either (\ref{block-sym}) or (\ref{block-asym}). In the case where we consider recoloring single edges, the coloring vector $V$ has length $\binom{n}{2}$. In the algorithms described below, we use the notation $V_{i,c}$ to indicate the vector obtained from a coloring vector $V$ by assigning color $c$ to component $i$ of vector $V$. For a given coloring vector $V$, let $f_{c}(V)$ denote the number of monochromatic complete graphs of order $k_{c}$ in color $c$. During the search, the goal is to minimize $f_{c}$ for each $ 1 \leq c \leq r$; but attempting to minimize a simple count of bad subgraphs does not appear to be the best strategy. So we adjust our objective function by defining the $score$ of a coloring vector $V$ as a weighted sum \[ score(V) = \sum_{c=1}^{r} w_{c} \cdot f_{c}(V) \] \noindent where $w$ is a scalar array of {\em weights} adjusted during the course of the algorithm. The array $w$ is useful for two reasons. The first reason applies to off-diagonal Ramsey numbers and deals with the relative difficulty of eliminating badgraphs in the two colors. Consider the case of an off diagonal Ramsey number, $R(s,t)$, where $s < t$. One will find that it is easier to eliminate the monochromatic $K_t$ subgraphs than to eliminate the monochromatic $K_s$ subgraphs. Local search procedures will tend to generate colorings with a large number of bad $K_s$ subgraphs and few (or no) bad $K_t$ subgraphs. This tendency increases as $|s - t|$ increases. To combat this problem, one can use weights such that $w_{s} > w_{t}$. A good rule of thumb is to begin the search procedure with weights that satisfy \[ \frac{t}{s} \leq \frac{w_{s}}{w_{t}} \leq \Big(\frac{t}{s}\Big)^2. \] A second reason for using weights is that they help avoid getting stuck in local minima. As the search proceeds, we adjust the weights so they are proportional to a moving average of the bad subgraph counts. For the problems in this paper, we updated $w$ for each color $c$ by \[ w_c = \frac{K w_c + f_c}{(K+1)\sum_{i=1}^{r} f_i} \] \noindent where $K$ is a positive constant. The choice of $K$ seems important. When $K$ is too small, the weights will oscillate wildly, and so will the number of bad subgraphs. On the other hand, if $K$ is too large, the weights will change too slowly, and the algorithm is susceptible to getting trapped in local minima. Values of $K$ between $10$ and $100$ seem to be effective for the problems considered in this paper. In addition, the weights can be changed randomly. The relative size of the random contribution to the weights is analogous to the temperature variable in simulated annealing \cite{metro}. In the discussions below, we have occasion to mention the CPU time required to obtain a certain bound. All such times are approximate times needed for a single CPU core, running at $3.0$ gHz (an average). Several bounds obtained here required months of CPU time. In these cases, the work was performed using $288$ CPU cores at the Department of Mathematics and Computer Science at the Indiana State University. The majority of these CPUs are third generation Intel {\em i5}'s. With this setup, a month of CPU time can be covered in less than three hours. \subsection{Method 1} Our basic search method combines steepest descent with tabu search \cite{tabu}. An outline of the search procedure is displayed in the Algorithm 1. We emphasize that the coloring vector $V$ can be any one of the three types mentioned above: circulant, block-circulant, or individual edges. The procedure begins with a random coloring vector (which determines the adjacency coloring matrix) and maintains a record of the last $L$ colorings. During each iteration, the procedure considers each of the possible recoloring vectors $V_{i,c}$ not in the tabu list. A recoloring vector that produces the minimum score is chosen and applied. If $L$ is large relative to $n$, it is possible for the algorithm to enter a state in which no recoloring $V_{i,c}$ can be used without revisiting a state on the tabu list. If this (rare) event occurs, we restart with a new random coloring. When coloring graphs of order $n$, where $150 < n < 200$ (which accounts for several of the cases considered here), setting $L = 1000$ seemed to produce good results. \vspace{3mm} \begin{algorithm}[H] \caption{Steepest descent with tabu search} \vspace{1mm} \KwIn{Coloring vector $V$, number of colors $r$, size of a tabu list $L$} \Begin{ Set elements of $V$ to random colors\; $H \gets \{\}$\; \Repeat{$ score(V)=0 $ {\bf or} $ M=\varnothing $} { $W \gets \{\}$\; \For{$1 \leq i \leq \left\vert V \right\vert $}{ \For{$1 \leq c \leq r$}{ \If{$V_{i, c} \not\in H$} { $W = W \cup V_{i, c}$\; Compute $ score(V_{i, c}) $\; } } } $M \gets \{V \in W \,|\, score(V) \mbox{ is a minimum}\}$\; \If{ $ M \ne \varnothing $ } { Randomly choose new $V \in M$\; $H \gets H \cup V$\; \If{$\left\vert H \right\vert > L$} { Remove an element from $H$ in FILO manner\; } Adjust weights\; } } \Return{$V$}\; } \end{algorithm} \vspace{3mm} \subsection{Method 2} In addition to the weight based method given in Algorithm 1, we also used a procedure based on simulated annealing \cite{metro}. This method is not suitable for circulants, but performed well for several block-circulant colorings, and is a good alternative when the recoloring of the individual edges is considered. This local improvement search procedure is outlined in Algorithm 2. At the beginning of the procedure, we find the set of all edges contained in bad subgraphs of order $k_{c}$ in any of the colors $1\leq c \leq r$. This edge set is denoted by $I$ and can be created during the bad subgraph counting procedure, using very little additional CPU time. In the inner loop of Algorithm 2, we apply simulated annealing on edges from the set $I$. The initial temperature $T_{0}$ and the cooling rate depends on $n$ (the order of the graph) and the initial $score$. The inner loop is limited by $j_{max}$ iterations, after which we update the set $I$. Typically we would set $j_{max} = \left\vert V \right\vert / \, 4$. In most cases when $T$ reached $0$, the coloring can be improved further, so we restarted the procedure using the newly obtained coloring vector $V$. The process is repeated until the further improvements can not be made. To perform a circulant block search, several modification are required. Initially $V$ is set to a random coloring vector. Next, we set $I = \{1,\dots,\left\vert V \right\vert\}$. The optimal choice of the initial temperature $T_{0}$ and cooling rate varies greatly for different $n$, $m$ and $r$, and we attempt to determine them empirically. In practice, if $score(V)$ is decreasing more slowly than expected, we restart the entire procedure using the same initial parameters. \vspace{3mm} \begin{algorithm}[H] \caption{Search procedure based on simulated annealing} \vspace{1mm} \KwIn{Coloring vector $V$, number of colors $r$, initial temperature $T_{0}$, number of iterations of the inner loop $j_{max}$} \Begin{ $T \gets T_{0}$\; $s\gets \infty$\; \Repeat{$ s=0 $ {\bf or} $T = 0$} { $I \gets \{\}$\; \For{$1 \leq i \leq \left\vert V \right\vert $}{ $c \gets V[i]$\; \If{$f_c(V_{i,c+1}) < f_c(V)$}{ $I \gets I \cup \{i\}$\; } } \For{$1 \leq j \leq j_{max} $}{ Randomly choose $i \in I$\; Randomly choose $c \in \{1,\dots,r\}$, $c \ne V[i]$\; $s^{*} \gets score(V_{i,c})$\; \If{$s^{*} - s \leq 0$ \bf{or} $e^{(s - s^{*})/T} > random(0,1)$}{ $V \gets V_{i,c}$\; $s \gets s^{*}$\; } } Reduce $T$\; Adjust weights\; } \Return{$V$}\; } \end{algorithm} \subsection{Special Constructions} Several lower bounds presented in this paper required a combination of our basic search methods. A brief summary of how these colorings were obtained is given below. \subsubsection{$R(5,10)$} The lower bound for $R(5,10)$ (labeled {\it circulant-minus} in Table 1) required two steps. First, we began with a circulant coloring of $K_{149}$ that was nearly a good coloring. This particular coloring had $149$ monochromatic $K_5$'s and no monochromatic $K_{10}$'s. After we had applied a local search procedure, one problem vertex was deleted, and a local search for a good coloring succeeded. \subsubsection{$R(4,8)$} To improve the lower bound on $R(4,8)$, we examined colorings of $K_n$ for $60 \leq n \leq 64$, and for all $m$ such that $m\mid n$ and $2 \leq m \leq 10$. The circulant block search was performed using the simulated annealing procedure. We then performed local searchs on those graphs we found with fewer than $250$ bad subgraphs. For $m = 4$ and $m = 5$, graphs on $60$ vertices were found and used to extract a $(4,8)$-coloring on $58$ vertices. In the first case, the block construction contained $60$ monochromatic $K_4$'s and $45$ monochromatic $K_{8}$'s. In the second, it contained $63$ monochromatic $K_4$'s and no monochromatic $K_{8}$'s. In both cases, the best coloring was reached after removing two vertices. We note that these graphs on $60$ vertices were not those with the minimum possible number of bad subgraphs (among block circulant colorings). Many colorings were found with fewer than $100$ bad subgraphs, and some with fewer than $50$, but we were not be able to modify these to obtain good colorings. To obtain a new $(4,8)-$coloring on $K_{58}$ vertices, several hours of CPU time were required. We spent over a month of CPU time trying to improve this solution. We found several close graphs on $K_{59}$ and $K_{60}$. On $K_{59}$, we were able to find a coloring with only $3$ monochromatic $K_4$'s and no monochromatic $K_{8}$'s. \subsubsection{$R(4,11)$} Perhaps our most complicated search was for the case of $R(4,11)$. This search required four steps, beginning with a $(4,8)$-coloring, extending it to a $(4,9)$-coloring, then to a $(4,10)$-coloring, and finally to a $(4,11)$-coloring. We started from a circulant block construction for $n=60$ and $m=4$, the same one we had used for $R(4,8)$. We extended this graph by appending circulant $15\!\times\!15$-blocks to the $60 \times 60$ coloring, and without changing the coloring on the first $60$ vertices. A promising $(4,9)$-coloring on $75$ vertices was found. Next we added another layer of $15\times15$-blocks and obtained an even more promising $(4,10)$-coloring with $n=90$ and $m=6$. Then we extended this coloring to a (nearly good) $(4,11)$-coloring on $105$ vertices. After repeatedly reaching local minima, we became convinced that we were not going to find a good coloring while maintaining the circulant block structure. At that point we removed four vertices and applied a local search procedure recoloring individual edges, and obtained a good coloring on $101$ vertices, thus establishing the new lower bound for $R(4,11)$. The promising coloring on $105$ vertices required several hours of CPU time to locate. The final local search required approximately a week of CPU time. \subsubsection{$R(3,3,6)$} A graph that we find interesting was uncovered when applying a circulant block search to the three color Ramsey number $R(3,3,6)$. The best result was obtained for $n=60$ and $m=6$, and no additional corrections were required. The coloring has significant symmetry. The graph in the first color has $2^{23} \cdot 5 = 41943040$ automorphisms; the color two graph $80$ automorphisms; and the color three graph has $40$ automorphisms. This coloring was discovered using the procedure similar to the one described in Algorithm 1, and the search took only a few minutes of CPU time. It is interesting that several other search procedures we tried were not able to find a good coloring on $K_{60}$. \subsection{Using Special Colorings} In certain cases, when two Ramsey numbers $R(s,t)$ and $R(s,t+k)$ are close, we tried to obtain a new lower bound for $R(s,t + k)$ by using the coloring that established the lower bound for $R(s,t)$ as a starting point for the $(s,t+k)$ search. Copies of $(s,t)$-coloring were used as starting blocks in a block circulant coloring that we hoped to manipulate into an $(s,t+k)$-coloring. If the order of the desired coloring was not an integer multiple of the order of the good $(s,t)$ coloring, we deleted rows and columns until we had a matrix of the desired size. If the resulting graph had only a few bad subgraphs, we tried to improve it either by using a block circulant search or a single edge recoloring procedure. \subsubsection{Colorings derived from Paley coloring of $K_{101}$} To obtain the new bounds for $R(6,7)$ and $R(6,8)$ we used the Paley coloring of $K_{101}$, which was the graph used to establish the longstanding lower bound for $R(6,6)$ \cite{kalb}. In both cases, the initial block coloring created from $101 \times 101$ circulant blocks was reduced to a matrix of the desired size, and the resulting coloring contained a relatively large high number of bad subgraphs. By applying the block circulant search procedure, we reduced the number of bad subgraphs significantly. To obtain good colorings, several days of CPU time were required. Additionally, several months of CPU time were spent trying to further improve the bounds. During this final stage, both bounds were improved by one. Let us note that a similar result can be obtained for $R(6,7)$ by using only the single edge recoloring procedure. but the search required more CPU time. For $R(6,8)$, it was necessary to make improvements in the block circulant structure first, and then apply single edge recoloring. \subsubsection{Colorings derived from a cubic coloring of $K_{127}$} The new lower bounds for $R(4,t)$ for $13\leq t \leq 16$ were obtained by using the cubic coloring of $K_{127}$. The cubic coloring was used to establish the lower bound for R(4,12) \cite{sll}. The previous bound for $R(4,13)$ was only $5$ greater than that for $R(4,12)$, and in the case of $R(4,14)$ the difference was only $13$. It was expected that the search would succeed at least for $t=13$, but surprisingly, we quickly obtained the other good colorings. The initial block coloring created from copies of the cubic residue graph on $K_{127}$ contained only few bad subgraphs. However, performing a local search on those colorings was particularly slow. In all four cases, we feel that further improvement is possible. But better bounds would likely require years of CPU time using this method. \subsection{Splitting} This section describes a simple method that allowed us to reach most of the lower bounds given in Table 2. The idea is to use known $(R(s,t)-u,k)$-colorings to create $(s,t,k)$-colorings, for some $u\ge 1$. We noticed that some of the bounds for Ramsey numbers of the form $R(3,3,k)$ were close to the corresponding bounds for numbers of the form $R(5,k)$. A similar observation was made with regard to $R(3,4,k)$ and $R(7,k)$. In the first case, $5$ is just one less than the Ramsey number $R(3,3)$, and in the second case $7$ is two less than $R(3,4)$. So it appeared conceivable that we could use known $(5,k)$-colorings to obtain new $(3,3,k)$-colorings, and similarly use $(7,k)$-colorings to obtain new $(3,4,k)$-colorings. In the first case, we have to split the first color into two $K_3$-free graphs. In the second case, we have to split the first color into a $K_3$-free graph and a $K_4$-free graph. As a starting point we used $2$-color circulant colorings obtained by the method described in Section 3.1. First, we attempt to split the color one graph into two circulants. If good coloring can not be achieved, we continue the process by recoloring individual edges. At this point, we still do not modify the third color. In most of the cases, we found good colorings almost instantly. Of course there is no guarantee that this procedure will work in general. In some cases, we were not able to find a good coloring using a given circulant, so we generated other non-isomorphic circulant colorings and tried each of them. Usually after few attempts the procedure succeeded. In several cases, we were unable to find good circulant colorings. But if the number of bad subgraphs was small enough, we ran the single edge recoloring procedure again, this time allowing changes to the third color. We were thereby able to find, for example, a good $(3,3,12)$-coloring on $193$ vertices. But the process took a few hours of CPU time. Note that the lower bounds we found for $R(3,3,13)$ and $R(3,3,16)$ are slightly less than those found for $R(5,13)$ and $R(5,16)$, reflective of the fact that we could not complete the splitting procedure for our best $(5,13)$ and $(5,16)$ colorings, but did succeed using smaller colorings. \section{Conclusions} One of the motivations of our work was to apply a fixed amount of resources, both computer and human, to a range of Ramsey problems, and learn to what extent the lower bounds could be improved. We focused on classical two-color Ramsey numbers where the best known lower bound was between $100$ and $200$. We did this for two reasons. First, neither of us had ever applied serious effort to problems in this range, and second, we felt these would be problems where progress could be made. For problems where the current lower bound is less than $100$, we felt that it was unlikely that progress would be made by simple methods (the case of $R(4,8)$ turned out to be an exception). For problems where the current lower bound is significantly greater than $200$, we felt that the required computer time would usually be too great. In each of the cases considered, we applied all of the methods described here, and few others. Other methods tried, but not mentioned here include the use of general (i.e., non-cyclic) Cayley colorings. It is remarkable that for every problem considered here, circulant colorings do better (produce good colorings for a larger value of $n$) than Cayley colorings from other groups. One obvious explanation for this is that cyclic groups have elements of larger average order than other groups, but it is still somewhat surprising. Finally we considered the case of $R(8,8)$, where the current lower bound seems much too small, as one may observe from the Figure (the data is taken from \cite{survey}). Despite signifcant computer time spent searching for circulant colorings in the $282$ to $286$ range, we did not find anything new. It is tempting to conclude that if there is a circulant coloring that improves the lower bound for $R(8,8)$, it is quite a bit larger than the current best coloring. In fact, even the related case of $R(7,8)$ is surprisingly difficult, especially considering that the current bound of $217$ is only $12$ more than the $R(7,7)$ bound. We applied a variety of methods to this problem with no success. \vspace{5mm} \begin{figure}[H] \centering \begin{tikzpicture}[scale=0.7] \draw (2,1) -- coordinate (x axis mid) (11,1); \draw (2,1) -- coordinate (y axis mid) (2,10.2); \foreach \x in {3,...,10} \draw (\x,1.05) -- (\x,0.9) node[anchor=north] {\x}; \foreach \y in {2,...,10} \draw (2.05,\y) -- (1.9,\y) node[anchor=east] {\y}; \node[below=0.75cm] at (x axis mid) {$n$}; \node[rotate=90, above=0.75cm] at (y axis mid) {$log_2 R(n,n)$ \, (lower bound)}; \draw plot[mark=*,mark options={fill=blue}] file{log.data}; \draw[->,thick,red,>=stealth] (9,6) node[sloped,above,right] {R(8,8)} -- (8.06,8); \end{tikzpicture} \caption{Log plot of current lower bounds for $R(n,n)$} \end{figure} \bibliographystyle{plain} \begin{thebibliography}{99} \bibitem{tabu} F. Glover, \newblock Tabu Search -- Part 1, \newblock {\em ORSA Journal on Computing}, 1: 190--206, 1990. \bibitem{HaKr} H. Harborth and S. Krause, \newblock Ramsey Numbers for Circulant Colorings, \newblock {\em Congressus Numerantium}, 161: 139--150, 2003. \bibitem{hillirving} R. Hill and R.W. Irving, \newblock On Group Partitions Associated with Lower Bounds for Symmetric Ramsey Numbers, \newblock {\em European Journal of Combinatorics}, 3: 35--50, 1982. \bibitem{kalb} J.G. Kalbfleisch, \newblock Construction of Special Edge-Chromatic Graphs, \newblock {\em Canadian Mathematical Bulletin} 8: 575--584, 1965. \bibitem{metro} S. Kirkpatrick, C. D. Gelatt and M. P. Vecchi, \newblock Optimization by Simulated Annealing, \newblock {\em Science}, 220: 671--680, 1983. \bibitem{survey} S. P. Radziszowski, \newblock Small Ramsey Numbers, \newblock {\em The Electronic Journal of Combinatorics}, \#DS1, 2014. \bibitem{xurad} X. Xu and S. P. Radziszowski, \newblock On Some Open Questions for Ramsey and Folkman Numbers \newblock {\em The Royal Swedish Academy of Sciences}, Report 19, 2013/2014. \bibitem{sll} S. Wenlong, L. Haipeng and L. Qiao, \newblock New Lower Bounds of Classical Ramsey Numbers R(4,12), R(5,11) and R(5,12), \newblock {\em Chinese Science Bulletin}, 43(6): 528, 1998. \end{thebibliography} \end{document}