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\begin{center}
\vskip 1cm{\LARGE\bf 
Some More Van der Waerden Numbers 
}
\vskip 1cm
\large
Tanbir Ahmed  \\
Department of Computer Science and Software Engineering \\
Concordia University \\
Montr\'eal, QC H3G1M8 \\
Canada \\
\href{mailto:ta\_ahmed@cs.concordia.ca}{\tt ta\_ahmed@cs.concordia.ca}\\
\end{center}

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\begin{abstract}
The \emph{van der Waerden number} $w(k; t_0, t_1,\ldots,t_{k-1})$ is
the smallest positive integer $n$ such that every $k$-coloring of the
sequence ${1,2,\ldots,n}$ yields a monochromatic arithmetic progression
of length $t_i$ for some color $i\in\set{0,1,\ldots,k-1}$.  In this
paper, we propose a problem-specific backtracking algorithm for
computing van der Waerden numbers $w(k;t_0,t_1,\ldots,t_{k-1})$ with
$t_0=t_1=\cdots=t_{j-1}=2$, where $k\geq j+2$, and $t_i\geq 3$ for
$i\geq j$.  We report some previously unknown van der Waerden numbers
using this method.  We also report the exact value of the previously
unknown van der Waerden number $w(2;5,7)$.
\end{abstract}

\section{Introduction}
The \emph{van der Waerden number} $w(k; t_0, t_1,\ldots,t_{k-1})$ is the smallest positive integer
$n$ such that every $k$-coloring of the sequence $1,2,\ldots,n$ yields a monochromatic
arithmetic progression of length $t_i$ for some color $i\in\set{0,1,\ldots,k-1}$.
For a list of all known values of $w(k; t_0, t_1,\ldots,t_{k-1})$ and corresponding references,
see Ahmed \cite{ahmed2009, ahmed2010, ahmed2011}, Ahmed, Kullmann, and Snevily \cite{aks2011}, and Kouril \cite{kouril2012}. 
A {\it good $k$-coloring} of the set $\set{1,2,\ldots,n}$
corresponding to $w(k; t_0,t_1,\ldots,t_{k-1})$ contains 
no monochromatic arithmetic progression of length $t_i$ for any $i$.
We call such a good $k$-coloring of $1,2,\ldots,n$ a \emph{certificate} of the 
lower bound $w(k;t_0,t_1,\ldots,t_{k-1})&gt;n$.
We denote colorings as strings;
for example, 00110011 means the color partition $\{1,2,5,6\}\cup\{3,4,7,8\}$.

In this paper, we propose a problem-specific backtracking algorithm for computing 
van der Waerden numbers $w(k;t_0,t_1,\ldots,t_{k-1})$
with $t_0=t_1=\cdots=t_{j-1}=2$ where $k\geq j+2$, and $t_i\geq 3$ for $i\geq j$.
We report some previously unknown numbers using this method.

We also report the previously unknown value of $w(2; 5,7)$ to be $260$. 
By far, only two values in the sequence $w(2;5,t)$ for $t\geq 5$ are known, 
namely $w(2;5,5)=178$ (Stevens and Shantaram \cite{stevens1978})  and $w(2;5,6)=206$ (Kouril \cite{kouril2006}).

\section{Some new values of \texorpdfstring{$w(k; t_0,t_1,\ldots,t_{k-1})$}.}
In this section, we discuss an idea to compute van der Waerden numbers with
specific values of $t_0,t_1,\ldots,t_{k-1}$ taking symmetry into consideration. 

\subsection{On \texorpdfstring{$w(k; 2,2,\ldots,2, t_j,t_{j+1}\ldots,t_{k-1})$}.}\label{sec-j2s-a}
Suppose in $w(k; t_0,\ldots,t_{j-1},t_{j},\ldots,t_{k-1})$ where $k-j\geq 2$, we have 
$t_0=t_1=\cdots=t_{j-1}=2$, and
$t_i\geq 3$ for $i=j,j+1,\ldots,k-1$. Any certificate of a lower bound of this van der Waerden number will contain
each of $0,1,\ldots,j-1$ exactly once. Hence the certificate will still remain valid after any in-place permutation 
of $0,1,\ldots,j-1$ in the certificate. For example, $898998879898031546989829988989$ is a certificate of the lower bound 
$$w(10;2,2,2,2,2,2,2,2,3,3)&gt;30,$$
which uses 10 colors. Keeping $8$ and $9$ in place, there are $8!$ certificates that prove the same lower bound.

In such a case, any certificate containing $k$ colors can be transformed into an equivalent certificate 
replacing each of $0,1,\ldots,j-1$ with a symbol $x$, and keeping the remaining $k-j$ colors. 
When we extend a certificate, we prohibit $t_i$-term arithmetic progressions
for $i=j,j+1,\ldots,k-1$ and check that the number of $x$ does not exceed $j$. 
This observation greatly reduces the search space (the backtrack search-tree becomes $(k-j+1)$-ary instead of $k$-ary) 
of a trivial backtrack algorithm 
and makes way for computing new van der Waerden numbers. 

From the above discussion, an equivalent certificate in our example is $$8989988x9898xxxxxx9898x9988989,$$  
which uses only two colors and a symbol $x$. For computational convenience, we can write this
certificate as $$121221102121000000212102211212,$$
with symbol $x$ being replaced by integer color $0$ and color $c$ being replaced by integer color $c-j+1$.

\subsection{On \texorpdfstring{$w(k; 2,2,\ldots,2, t,t\ldots,t)$ with $t\geq 3$}.}\label{sec-j2s-b}
Let $t_0=t_1=\cdots=t_{j-1}=2$ and $t_i=t\geq 3$ for $i=j,j+1,\ldots,k-1$.
We can further minimize the backtrack search-space by extending only one
certificate from the set of isomorphic certificates under symmetry. 
Consider the $48$ certificates of the lower bound $w(3;3,3,3)&gt;26$
with the colors named $1$, $2$, and $3$.

\begin{table}[H]
\begin{center}
\begin{tabular}{|r|l|r|l|}
\hline 
1: &amp; 11221123233131121223133232 &amp; 2: &amp; 11223113132233223131132211 \\   
3: &amp; 11232113132233223131123211 &amp; 4: &amp; 11323112123322332121132311 \\   
5: &amp; 11331132322121131332122323 &amp; 6: &amp; 11332112123322332121123311 \\   
7: &amp; 12112123322332121123311313 &amp; 8: &amp; 12122113223231133113232231 \\   
9: &amp; 12122113223231133113232232 &amp; 10: &amp; 12122321131332322121331133 \\   
11: &amp; 12332321122112323321133131 &amp; 12: &amp; 13113132233223131132211212 \\   
13: &amp; 13133112332321122112323321 &amp; 14: &amp; 13133112332321122112323323 \\   
15: &amp; 13133231121223233131221122 &amp; 16: &amp; 13223231133113232231122121 \\   
\hline
17: &amp; 21211223113132233223131131 &amp; 18: &amp; 21211223113132233223131132 \\   
19: &amp; 21211312232331311212332233 &amp; 20: &amp; 21221213311331212213322323 \\   
21: &amp; 21331312211221313312233232 &amp; 22: &amp; 22112213133232212113233131 \\   
23: &amp; 22113223231133113232231122 &amp; 24: &amp; 22131223231133113232213122 \\   
25: &amp; 22313221213311331212231322 &amp; 26: &amp; 22331221213311331212213322 \\   
27: &amp; 22332231311212232331211313 &amp; 28: &amp; 23113132233223131132211212 \\   
29: &amp; 23223231133113232231122121 &amp; 30: &amp; 23233132212113133232112211 \\   
31: &amp; 23233221331312211221313312 &amp; 32: &amp; 23233221331312211221313313 \\   
\hline
33: &amp; 31221213311331212213322323 &amp; 34: &amp; 31311213323221211313223322 \\   
35: &amp; 31311332112123322332121121 &amp; 36: &amp; 31311332112123322332121123 \\   
37: &amp; 31331312211221313312233232 &amp; 38: &amp; 32112123322332121123311313 \\   
39: &amp; 32322123313112122323113311 &amp; 40: &amp; 32322331221213311331212212 \\   
41: &amp; 32322331221213311331212213 &amp; 42: &amp; 32332321122112323321133131 \\   
43: &amp; 33112332321122112323321133 &amp; 44: &amp; 33113312122323313112322121 \\   
45: &amp; 33121332321122112323312133 &amp; 46: &amp; 33212331312211221313321233 \\   
47: &amp; 33221331312211221313312233 &amp; 48: &amp; 33223321211313323221311212 \\   
\hline
\end{tabular}
\end{center}
\caption{All certificates of $w(3;3,3,3)&gt;26$}
\label{tab-333-cert}
\end{table}


Let a permutation $\pi$ of $1,2,\ldots,k$ be a sequence $\pi(1),\pi(2),\ldots,\pi(k)$.
Let $S(k)$ denote the set of all permutations of $1,2,\ldots,k$. We write the permutations
in $S(k)$ in parenthesized notation with respect to the indices $1,2,\ldots,k$. For example,
$$S(3)=\set{(1)(2)(3), (1)(2,3), (1,2)(3), (1,2,3), (1,3,2), (1,3)(2)}.$$

Let $C=c_1c_2\cdots{c_n}$ denote a certificate of the lower bound $w(k; t,t,\ldots,t)&gt;n$. 
Define $T_\pi(C)$ and $T_{S(k)}(C)$ by $\pi(c_1)\pi(c_2)\ldots\pi(c_n)$
and $\set{T_{\pi}(C) : \pi\in S(k)}$, respectively.

For example, $T_{S(3)}(11221123233131121223133232)$ equals the set with the following elements
\begin{center}
$\begin{array}{ll}
11221123233131121223133232, &amp; 11331132322121131332122323, \\
22112213133232212113233131, &amp; 22332231311212232331211313, \\
33113312122323313112322121, &amp; 33223321211313323221311212. \\
\end{array}$
\end{center}

Similarly, all $48$ certificates can be generated from the following $8$ certificates:

\begin{longtable}{|r|l|r|l|}
\caption[Representative certificates of $w(3;3,3,3)&gt;26$]{Representative certificates of $w(3;3,3,3)&gt;26$}\label{tab-333-rep-cert}\\
\endlastfoot
\hline
1: &amp; 11221123233131121223133232 &amp; 2: &amp; 11223113132233223131132211 \\   
3: &amp; 11232113132233223131123211 &amp; 7: &amp; 12112123322332121123311313 \\   
8: &amp; 12122113223231133113232231 &amp; 9: &amp; 12122113223231133113232232 \\   
10: &amp; 12122321131332322121331133 &amp; 11: &amp; 12332321122112323321133131 \\   
\hline
\end{longtable}

So instead of generating and extending all certificates, we can consider
only one from the $3!$ equivalent certificates.
To do so, we can observe that, in a certificate $c_1c_2\cdots{c_n}$ of $w(k; t,t,\ldots,t)&gt;n$,
if $c_i$ is greater than $c_\ell$ for $1\leq \ell\leq i-1$, then we can ignore branching on
$c_i+1, c_i+2,\ldots,k$ at position $i$. 

\subsection{The algorithm}
We combine the ideas from Sections \ref{sec-j2s-a} and \ref{sec-j2s-b} 
to obtain the following algorithm 
for $w(k;t_0,t_1,\ldots,t_{k-1})$, where $t_0=t_1=\cdots=t_{j-1}=2$ and $k\geq j+2$.


\begin{algorithm}[ht]
\caption{\sc Recursive algorithm Run({$k$, $j$, $index$, $x$})}
\begin{algorithmic}[1]
\Function{Run}{$k$, $j$, $index$, $x$}
\State {\bf if $zeroCount&gt;j$ then return end if}
\If {$index&gt;0$ and $x&gt;0$}
   \If{the indices of $t_{x+j-1}$ $x$'s in $c_1c_2\cdots{c_{index}}$ form an AP} 
       \State {\bf return}
   \EndIf
\EndIf
\State {\bf if $index&gt;max$ then} $max = index$ {\bf end if}
\For{$i=0$ to $k-j$}
   \State {\bf if $i=0$ then} $zeroCount = zeroCount+1$ {\bf end if}
   \State $c_{index+1} = i$
   \State {\sc Run}($k$, $j$, $index+1$, $i$)
   \State {\bf if $i=0$ then} $zeroCount = zeroCount-1$ {\bf end if}
   \If{$i&gt;0$ and $t_j=t_{j+1}=\cdots=t_{k-1}=t$}
      \If {$index \leq j+(i-1)(t-1)+1$}
         \If {$c_{index+1}&gt;c_\ell$ for $1\leq\ell\leq index$}  
            \State {\bf break}
         \EndIf
      \EndIf
   \EndIf
\EndFor
\EndFunction
\end{algorithmic}
\label{alg-run} 
\end{algorithm}



We can observe that function {\sc Run} in Algorithm 1 returns with $$max+1 = w(k;2,2,\ldots,2,t_{j},t_{j+1},\ldots,t_{k-1})$$
when called as {\sc Run($k$,$j$,0,0)} with $zeroCount$ and $max$ initialized to zero.


\subsection{Experiment on some known van der Waerden numbers}
In Table \ref{tab-exp},
we report test-results of Algorithm 1 
with parameters corresponding to some known van der Waerden numbers. 
We consider numbers that are relevant to the algorithm and 
take less than half an hour of run-time. 

\begin{table}[ht]
\begin{center}
\begin{tabular}{|l|l|l|r|}
\hline
 &amp; $(t_j,t_{j+1},\ldots,t_{k-1})$ &amp; $max+1$ &amp; $time (s)$\\
\hline\hline
{\sc Run($2$, $0$, $0$, $0$)} &amp; ($3$,$3$) &amp; $9 = w(2;3,3)$ &amp; $0.00$ \\
{\sc Run($2$, $0$, $0$, $0$)} &amp; ($4$,$4$) &amp; $35 = w(2;4,4)$ &amp; $0.00$ \\
{\sc Run($3$, $1$, $0$, $0$)} &amp; ($3$,$3$) &amp; $14 = w(3;2,3,3)$ &amp; $0.00$ \\
{\sc Run($3$, $1$, $0$, $0$)} &amp; ($4$,$4$) &amp; $40 = w(3;2,4,4)$ &amp; $0.38$ \\
{\sc Run($3$, $0$, $0$, $0$)} &amp; ($3$,$3$,$3$) &amp; $27 = w(3;3,3,3)$ &amp; $0.12$ \\
{\sc Run($4$, $2$, $0$, $0$)} &amp; ($3$,$3$) &amp; $17 = w(4;2,2,3,3)$ &amp; $0.00$ \\
{\sc Run($4$, $2$, $0$, $0$)} &amp; ($3$,$4$) &amp; $25 = w(4;2,2,3,4)$ &amp; $0.07$ \\
{\sc Run($4$, $2$, $0$, $0$)} &amp; ($3$,$5$) &amp; $43 = w(4;2,2,3,5)$ &amp; $2.20$ \\
{\sc Run($4$, $2$, $0$, $0$)} &amp; ($3$,$6$) &amp; $48 = w(4;2,2,3,6)$ &amp; $42.93$ \\
{\sc Run($4$, $2$, $0$, $0$)} &amp; ($4$,$4$) &amp; $53 = w(4;2,2,4,4)$ &amp; $10.25$ \\
{\sc Run($4$, $1$, $0$, $0$)} &amp; ($3$,$3$,$3$) &amp; $40 = w(4;2,3,3,3)$ &amp; $4.97$ \\
{\sc Run($5$, $3$, $0$, $0$)} &amp; ($3$,$3$) &amp; $20 = w(5;2,2,2,3,3)$ &amp; $0.00$ \\
{\sc Run($5$, $3$, $0$, $0$)} &amp; ($3$,$4$) &amp; $29 = w(5;2,2,2,3,4)$ &amp; $0.84$ \\
{\sc Run($5$, $3$, $0$, $0$)} &amp; ($3$,$5$) &amp; $44 = w(5;2,2,2,3,5)$ &amp; $38.11$ \\
{\sc Run($5$, $3$, $0$, $0$)} &amp; ($4$,$4$) &amp; $54 = w(5;2,2,2,4,4)$ &amp; $208.74$ \\
{\sc Run($5$, $2$, $0$, $0$)} &amp; ($3$,$3$,$3$) &amp; $41 = w(5;2,2,3,3,3)$ &amp; $102.71$ \\
{\sc Run($6$, $4$, $0$, $0$)} &amp; ($3$,$3$) &amp; $21 = w(6;2,2,2,2,3,3)$ &amp; $0.05$ \\
{\sc Run($6$, $4$, $0$, $0$)} &amp; ($3$,$4$) &amp; $33 = w(6;2,2,2,2,3,4)$ &amp; $7.66$ \\
{\sc Run($6$, $4$, $0$, $0$)} &amp; ($3$,$5$) &amp; $50 = w(6;2,2,2,2,3,5)$ &amp; $522.64$ \\
{\sc Run($6$, $3$, $0$, $0$)} &amp; ($3$,$3$,$3$) &amp; $42 = w(6;2,2,2,3,3,3)$ &amp; $1615.73$ \\
{\sc Run($7$, $5$, $0$, $0$)} &amp; ($3$,$3$) &amp; $24 = w(7;2,2,2,2,2,3,3)$ &amp; $0.31$ \\
{\sc Run($7$, $5$, $0$, $0$)} &amp; ($3$,$4$) &amp; $36 = w(7;2,2,2,2,2,3,4)$ &amp; $59.64$ \\
{\sc Run($8$, $6$, $0$, $0$)} &amp; ($3$,$3$) &amp; $25 = w(8;2,2,2,2,2,2,3,3)$ &amp; $1.38$ \\
{\sc Run($8$, $6$, $0$, $0$)} &amp; ($3$,$4$) &amp; $40 = w(8;2,2,2,2,2,2,3,4)$ &amp; $434.12$ \\
{\sc Run($9$, $7$, $0$, $0$)} &amp; ($3$,$3$) &amp; $28 = w(9;2,2,2,2,2,2,2,3,3)$ &amp; $5.58$ \\
\hline
\end{tabular}
\end{center}
\caption{Experiment on some known values}
\label{tab-exp}
\end{table}

\subsection{New values of \texorpdfstring{$w(k; t_0,t_1,\ldots,t_{k-1})$}.}
We have computed the following new values of $w(k; t_0,t_1,\ldots,t_{k-1})$ using Algorithm 1. 

\begin{longtable}{|l|l|}
\caption[New values of $w(k; t_0,t_1,\ldots,t_{k-1})$]{New values of $w(k; t_0,t_1,\ldots,t_{k-1})$}\label{tab-new}\\
\endlastfoot
\hline
$w(k; t_0,t_1,\ldots,t_{k-1})$ &amp; $$\\
\hline\hline
$w(7;2,2,2,2,2,3,6)$ &amp; $= 65$\\
\hline
$w(7;2,2,2,2,2,4,4)$ &amp; $= 66$\\
\hline
$w(7;2,2,2,2,3,3,3)$ &amp; $= 45$\\
\hline
$w(8;2,2,2,2,2,2,3,5)$ &amp; $= 61$\\
\hline
$w(8;2,2,2,2,2,2,3,6)$ &amp; $= 71$\\
\hline
$w(8;2,2,2,2,2,2,4,4)$ &amp; $= 67$\\
\hline
$w(8;2,2,2,2,2,3,3,3)$ &amp; $= 49$\\
\hline
$w(9;2,2,2,2,2,2,2,3,4)$ &amp; $= 42$\\
\hline
$w(9;2,2,2,2,2,2,2,3,5)$ &amp; $= 65$\\
\hline
$w(9;2,2,2,2,2,2,3,3,3)$ &amp; $= 52$\\
\hline
$w(10;2,2,2,2,2,2,2,2,3,3)$ &amp; $= 31$\\
\hline
$w(10;2,2,2,2,2,2,2,2,3,4)$ &amp; $= 45$\\
\hline
$w(10;2,2,2,2,2,2,2,2,3,5)$ &amp; $= 70$\\
\hline
$w(11;2,2,2,2,2,2,2,2,2,3,3)$ &amp; $= 33$\\
\hline
$w(11;2,2,2,2,2,2,2,2,2,3,4)$ &amp; $= 48$\\
\hline
$w(12;2,2,2,2,2,2,2,2,2,2,3,3)$ &amp; $= 35$\\
\hline
$w(12;2,2,2,2,2,2,2,2,2,2,3,4)$ &amp; $= 52$\\
\hline
$w(13;2,2,2,2,2,2,2,2,2,2,2,3,3)$ &amp; $= 37$\\
\hline
$w(13;2,2,2,2,2,2,2,2,2,2,2,3,4)$ &amp; $= 55$\\
\hline
$w(14;2,2,2,2,2,2,2,2,2,2,2,2,3,3)$ &amp; $= 39$\\
\hline
$w(15;2,2,2,2,2,2,2,2,2,2,2,2,2,3,3)$ &amp; $= 42$\\
\hline
$w(16;2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3)$ &amp; $= 44$\\
\hline
$w(17;2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3)$ &amp; $= 46$\\
\hline
$w(18;2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3)$ &amp; $= 48$\\
\hline
$w(19;2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3)$ &amp; $= 50$\\
\hline
$w(20;2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3)$ &amp; $= 51$\\
\hline
\end{longtable}

Based on the results in Table \ref{tab-new}, we have added and extended (as shown in bold fonts) the following entries in the OEIS: 
\begin{enumerate}
\item {\seqnum{A217005}:} $w(j+2; t_0,t_1,\ldots,t_{j-1},3,3)$ for $j\geq 0$
with $t_i=2$, $0\leq i\leq j-1$.
$$9,14,17,20,{21},{24},{25},{28},\tbf{31},\tbf{33},\tbf{35},\tbf{37},\tbf{39},\tbf{42},\tbf{44},\tbf{46},\tbf{48},\tbf{50},\tbf{51}.$$
\item {\seqnum{A217058}:} $w(j+2; t_0,t_1,\ldots,t_{j-1},3,4)$ for $j\geq 0$
with $t_i=2$, $0\leq i\leq j-1$.
$$18,21,25,29,33,36,40,\tbf{42},\tbf{45},\tbf{48},\tbf{52},\tbf{55}.$$
\item {\seqnum{A217059}:} $w(j+2; t_0,t_1,\ldots,t_{j-1},3,5)$ for $j\geq 0$
with $t_i=2$, $0\leq i\leq j-1$.
$$22,32,43,44,50,55,\tbf{61},\tbf{65},\tbf{70}.$$
\item {\seqnum{A217060}:} $w(j+2; t_0,t_1,\ldots,t_{j-1},3,6)$ for $j\geq 0$
with $t_i=2$, $0\leq i\leq j-1$.
$$32,40,48,56,60,\tbf{65},\tbf{71}.$$
\item {\seqnum{A217007}:} $w(j+2; t_0,t_1,\ldots,t_{j-1},4,4)$ for $j\geq 0$
with $t_i=2$, $0\leq i\leq j-1$.
$$35, 40, 53, {54}, {56}, \tbf{66}, \tbf{67}.$$
\item {\seqnum{A217008}:} $w(j+3; t_0,t_1,\ldots,t_{j-1},3,3,3)$ for $j\geq 0$
with $t_i=2$, $0\leq i\leq j-1$.
$$27, 40, 41, {42}, \tbf{45}, \tbf{49},\tbf{52}.$$
\end{enumerate}

\section{Exact value of \texorpdfstring{$w(2;5,7)$}. using SAT}
In this section, we report the exact value of $w(2;5,7)$ to be $260$. 

\subsection{\texorpdfstring{$w(2;5,7)\geq 260$}.}\label{sec-lb}
The following certificate (a good 2-coloring of $1,2,\ldots,259$)
establishes the lower bound $w(2;5,7)\geq 260$ (Ahmed \cite{ahmed2011}): 
\begin{center}

11111101 11101111 10000111 10000100 01110111 10100111 11001011 11011100 

01000010 11001001 10100001 00011101 11101001 11110010 11110111 00010000 

10110010 01101000 01000111 01111010 01111100 10111101 11000100 00101100 

10011010 a0010001 11011111 10011011 00101111 0111b011 00011011 01011110 

111. (ab is arbitrary)
\end{center}
\section{\texorpdfstring{$w(2;5,7)=260$}.}
It remains to show that every 2-coloring of $1,2,\ldots,260$ either contains a 5-term
arithmetic progression in color 0, or a 7-term arithmetic progression in color 1. 

\subsection{\texorpdfstring{$w(2;5,7)\leq 260$}.}
We construct an instance $F$ of the satisfiability problem (or SAT for short)  
with $260$ variables for the van der Waerden number $w(2;5,7)$ such that $F$ is satisfiable if and only if
$w(2;5,7)&gt;260$. For a brief introduction to SAT and SAT-encoding of van der Waerden numbers,
see Section 1 in Ahmed \cite{ahmed2010}.
We use a distributed application of an efficient implementation 
of the DPLL \cite{dpll} algorithm to show that the constructed instance is unsatisfiable. 
For a brief description of this implementation and its distributed application,
see Sections 3 and 4, respectively, in Ahmed \cite{ahmed2010}.

We have split the instance into 256 parts and then each of them into further parts
to distribute them over the cluster machines at Concordia. It took 200 2.2 GHz AMD Opteron processors to run roughly for 
a year to conclude that there is no good 2-coloring of $1,2,\ldots,260$ 
corresponding to $w(2;5,7)$.

In such a large computation where thousands of distributed branches of the search tree have run
on hundreds of processors, we hope we have not fallen into the trap of an
undetected hardware failure (an electricity failure is natural and every detected hardware-failure
 was re-run from the last state of the search), or a file-manipulation error
 on a particular branch which unfortunately could contain a good 2-coloring
of $1,2,\ldots,260$. We welcome interested readers with proper resources to
conduct another search to verify our result.

\section{Acknowledgements}
The author would like to thank Clement Lam for his continuous support,
Donald Knuth for his time and valuable suggestions, and  the anonymous referee 
for the helpful comments. The author would also like to thank Andalib Parvez for
carefully reading the manuscript.


\begin{thebibliography}{9}
\bibitem{ahmed2009}
\textrm{T. Ahmed}, Some new van der Waerden numbers and some van der
Waerden-type numbers, \emph{Integers}, {\bf 9} (2009), A06, 65--76.

\bibitem{ahmed2010}
\textrm{T. Ahmed}, Two new van der Waerden numbers: $w(2; 3, 17)$ and
$w(2; 3, 18)$, \emph{Integers}, {\bf 10} (2010), A32, 369--377.

\bibitem{ahmed2011}
\textrm{T. Ahmed}, On computation of exact van der Waerden numbers,
\emph{Integers}, {\bf 11} (2011), A71.

\bibitem{aks2011}
\textrm{T. Ahmed, O. Kullmann, and H. Snevily}, On the van der Waerden
numbers $w(2; 3, t)$, preprint, \url{http://arxiv.org/abs/1102.5433}.

\bibitem{dpll}  \textrm{M. Davis, G. Logemann, D. Loveland},
A machine program for theorem-proving,
\emph{Comm. ACM},
{{\bf 5}}
{(1962)},
{394--397}.

\bibitem{kouril2006} 
\textrm{M. Kouril}, 
A backtracking framework for Beowulf clusters with an extension to multi-cluster
computation and SAT benchmark problem implementation, 
\emph{Ph. D. Thesis, University of Cincinnati, Engineering : Computer Science and Engineering}, 2006.


\bibitem{kouril2012}
\textrm{M. Kouril}, \textrm{Computing the van der Waerden number $W(3,4)=293$},
\emph{Integers}, {\bf 12} (2012), A46. 

\bibitem{stevens1978} \textrm{R. Stevens and R. Shantaram},
Computer-generated van der Waerden partitions,
\emph{Math. Comp.},
{\bf 32} (1978), 635--636.

\bibitem{ubcsat} \textrm{D. A. D. Tompkins and H. H. Hoos}, UBCSAT: An
implementation and experimentation environment for SLS algorithms for
SAT and MAX-SAT. In Holger H. Hoos and David G. Mitchell, eds.,
{\it Theory and Applications of Satisfiability Testing of 2004}, Lecture
Notes in Computer Science, Vol.\ 3542, Springer, 2005, pp.\ 306--320.
\end{thebibliography}

\bigskip
\hrule
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\noindent 2010 {\it Mathematics Subject Classification}:
Primary 11B25; Secondary 05D10.

\noindent \emph{Keywords: }
van der Waerden number.

\bigskip
\hrule
\bigskip

\noindent (Concerned with sequences
\seqnum{A217005},
\seqnum{A217007}, 
\seqnum{A217008},
\seqnum{A217058},
\seqnum{A217059}, 
\seqnum{A217060}, and
\seqnum{A217037}.)


\bigskip
\hrule
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\vspace*{+.1in}
\noindent
Received September 28 2012;
revised version received  March 12 2013.
Published in {\it Journal of Integer Sequences}, March 16 2013.

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\noindent
Return to
\htmladdnormallink{Journal of Integer Sequences home page}{http://www.cs.uwaterloo.ca/journals/JIS/}.
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