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\markright{\sc the electronic journal of combinatorics 5 (1998), \#R7\hfill}
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\title{ All Ramsey numbers $r(K_{3},G)$ for connected graphs \\ of order 9 }

\author{ \small
\small Stephan Brandt \\
\small Fachbereich Mathematik \& Informatik, WE 2 \\
\small Freie Universit\"at Berlin \\
\small Arnimallee 3\\
\small 14195 Berlin, Germany \\
{\tt \small brandt@math.fu-berlin.de } 
\and
\small Gunnar Brinkmann \\
\small Fakult\"at f\"ur Mathematik\\ 
\small Universit\"at Bielefeld\\ \small 33501 Bielefeld, Germany \\
{\tt \small gunnar@mathematik.uni-bielefeld.de} 
\and 
\small Thomas Harmuth \\
\small Forschungsschwerpunkt Mathematisierung\\ 
\small Universit\"at Bielefeld\\ \small 33501 Bielefeld, Germany \\
{\tt \small harmuth@mathematik.uni-bielefeld.de} 
}

\date{\small Submitted: September 4, 1997; Accepted: January 3, 1998}

\maketitle


\begin{abstract}
We determine the Ramsey numbers $r(K_3,G)$ for all $261080$ connected
graphs of order $9$ and further Ramsey numbers of this type for some
graphs of order
up to $12$. Almost all of them were determined by computer programs
which are based on a program for generating maximal triangle-free graphs.
\end{abstract}


\section{Introduction}

\footnotetext[1]{AMS Subject Classification 05C55}

For two graphs $G$ and $H$, the Ramsey number $r(G,H)$ is the smallest
integer $r$, such that every $2$-colouring of $K_r$, using the colours 
red and blue, say, contains
$G$ as a red subgraph or $H$ as a blue subgraph. Equivalently, $r$ is
the smallest integer such that every graph $F$ of order $p\ge r$
contains $G$ as a subgraph, or its complement $\quer{F}$ contains $H$ as
a subgraph. The classical Ramsey numbers --- those where both $G$ and $H$
are complete graphs --- are notoriously difficult to compute or even to
estimate for large order graphs. Also
there are only a few precise results known for infinite sequences of
graphs. Usually only those cases are known where extremal graph theory and
Ramsey theory meet.

Here we will deal with Ramsey numbers $r(K_3,H)$ for connected graphs
$H$ of small order. A few years ago, these numbers were completely
determined only for graphs $H$ up to order $6$, the last (and major) step
being done by Faudree, Rousseau, and Schelp~\cite{FRS80}.
Only recently, the numbers for all connected graphs of
order $7$ were computed by Jin Xia~\cite{JinX93} in his thesis by using a
computer. Unfortunately, his results are unreliable since some of the
numbers turned out to be incorrect. The correct
numbers for all connected graphs of order $7$ and $8$ were computed by
Brinkmann~\cite{Bri96}, also using a computer. Independently,
Schelten and Schiermeyer computed the Ramsey numbers 
for graphs of order $7$ by hand  [15,16]. %\cite{SchSch96}\cite{SchSch96_2}.
We will give the Ramsey numbers for all connected graphs of order
$9$. The Ramsey numbers up to $r=27$ are determined by computer programs.
The remaining numbers are computed by Ramsey theoretical means or were
previously known. The computation of the Ramsey numbers
for {\em all} graphs of order $10$ seems currently out of reach, since
the upper and lower bounds for the Ramsey numbers $r(K_3,K_{10})$ and
$r(K_3,K_{10}-e)$, respectively, both differ by $3$ (see~\cite{Rad94}).

A second problem which we investigated is the concept of {\bf goodness}
introduced by Burr in~\cite{Burr81}. A connected graph $H$ is called
$G$-good, if the Ramsey number $r(G,H)$ is as small as possible for a
connected graph $H$, or, in other words,
\begin{equation} \label{e:good}
r(G,H) = (\chi (G) -1)(|H|-1) + s(G),
\end{equation}
where $\chi(G)$ denotes the chromatic number and
$s(G)$ the chromatic surplus of $G$, i.e.\ the minimum cardinality of a
color class taken over all proper $\chi(G)$ colorings of $G$. In the case
$G=K_p$ the most sparse connected graphs, namely trees, are known to
be $K_p$-good~\cite{Chv77}. In~\cite{BEFRS80} the question was raised
for the functions $f(K_p,n)$
and $g(K_p,n)$, where $f(K_p,n)$ is the largest integer such that every
connected graph of order $n$ with at most $f(K_p,n)$ edges is $K_p$-good and
$g(K_p,n)$ is the largest integer such that there is a $K_p$-good
connected graph
of order $n$ with $g(K_p,n)$ edges. We computed $f(K_3,n)$
and $g(K_3,n)$ for $n\le 12$.

Because of the large number of small Ramsey numbers computed here, 
trends for the asymptotic growth of Ramsey numbers can
be detected empirically. 
Some of them are opposed to the previously expected
behaviour. Motivated by our data, in~\cite{Bra96} the first author
succeeded in disproving several goodness conjectures for more general
classes of Ramsey numbers by purely theoretical (probabilistic) means. It
should be mentioned that the observed behaviour can hardly be detected
from the data on $r(K_3,H)$ for graphs $H$ of order at most $7$,
i.e.\ the range which can be solved without computer aid. We hope
that our data will serve to give further insight into the growth of
Ramsey numbers.

\section{Computing $r(K_3,H)$ for graphs of order 9}


We computed the Ramsey numbers $r(K_3,H)$ for connected graphs $H$ of
order $9$ (presented in the listings at the end of this paper) mostly
by using computer programs. The only exceptions are the cases
$r(K_3,K_9)=36$ and $r(K_3,K_9-e)=31$ (which are well-known --
see the regularly
updated dynamic survey of Radziszowski~\cite{Rad94}), 
$r(K_3,K_9-K_{1,s})=28$ for $2\le s\le 7$ and
$r(K_3,K_9-K_3) = 28$ which are determined with the help of standard
theoretical Ramsey arguments. The Ramsey number $r(K_3,K_n-K_{1,s})$
equals $r(K_3,K_{n-1})$ if $s$ is sufficiently large with respect to
$n$ and $r(K_3,K_{n-1})$, as we will show now.

Call a graph $F$ a {\em Ramsey graph}
for a pair of graphs $(G,H)$, if $F$ does not contain $G$ as a
subgraph and its complement $\quer{F}$ does not contain $H$ as a
subgraph. In order to show that $r(G,H)=r$, two tasks have to be
performed: (i) find a Ramsey graph of order $r-1$ for $(G,H)$, and
(ii) show that no Ramsey graph of order $r$ exists for $(G,H)$.
For (i) we can alternatively find a subgraph $H' \subseteq H$ with
$r(G,H')=r$.


\begin{thm} \label{t:complK1,s}
If $r=r(K_3,K_{n-1})$ and $n \ge s+1 > (n-2)(n-1)/(r-n)$ then $r(K_3,K_n-K_{1,s})=r$.
\end{thm}

\begin{pf}
Since $K_{n-1} \subseteq K_n-K_{1,s}$ we have $r(K_3,K_n-K_{1,s})\ge
r$. Now take a triangle-free graph $F$ of order $r$. We have to show
that $K_n-K_{1,s}$ is contained in the complement
$\quer{F}$. If the maximum degree $\Delta(F) \ge n$ then $\quer{F}$
contains $K_n$ since $F$ is triangle-free 
and therefore $K_n-K_{1,s}$.
So assume $\Delta(F) \le n-1$. By the definition of the Ramsey number,
$F$ must have an independent set $S$ of $n-1$
vertices. If $\Delta(F) = n-1$ choose $S$ to be the neighbourhood of
an $(n-1)$-valent vertex $v$, otherwise choose $v$ arbitrarily in
$V(F)\setminus S$. 
In any case, each vertex in $S$ has at
most $n-2$ neighbours in $V(F)\setminus (S\cup \{v\})$, so one vertex $w$
of the $r-n$ vertices in $V(F)\setminus (S\cup \{v\})$ has at most
$\lfloor (n-2)(n-1)/(r-n) \rfloor \le s$ neighbours in $S$. Hence
$K_n-K_{1,s}$ is contained in the subgraph of $\quer{F}$ induced by $S
\cup \{w\}$.
\end{pf}

Recently, Kim proved that $r(K_3,K_n) = \Theta (n^2/\log n)$~\cite{Kim95},
so Theorem~\ref{t:complK1,s} yields\\ $r(K_3,K_n-K_{1,s})=r(K_3,K_{n-1})$
if $s=\Omega(\log n)$. In the case that we are mainly interested in ($n=9$),
Theorem~\ref{t:complK1,s} gives equality for $s\ge 2$.

\begin{coro} \label{c:complK1,s}
$r(K_3,K_9-K_{1,s}) = 28$ for $2\le s\le 8$. \qed
\end{coro}

Since the complement of a triangle-free graph
contains $K_n-K_3$ if and only if it contains $K_n-K_{1,2}$,
Corollary~\ref{c:complK1,s} implies the following result.

\begin{coro}
$r(K_3,K_9-K_3) = 28$. \qed
\end{coro}




\subsection*{The algorithm}

The central tool for the computation of the remaining $261071$ Ramsey
numbers $r(K_3,H)$ for connected graphs $H$ of order $9$ and the further
Ramsey numbers which we computed, is the
computer program \mtf{} described in~\cite{BBH97}.
This program is designed to generate all non-isomorphic maximal
triangle-free graphs,
but it is prepared to include certain restrictions into the generation
process. One of these restrictions is to generate only Ramsey graphs
for $(K_3,H)$.


The program \mtf{} generates maximal triangle-free graphs $F$ on $n$
vertices from maximal triangle-free graphs 
$F'$ on $n-1$ vertices in such a way that $\quer{F'} \subseteq
\quer{F}$ always holds.
So if a graph $H$ is contained in the complement of a maximal triangle-free
graph $F_0$, it will be contained in the complement of all its
descendants as well and therefore they cannot be a Ramsey graph for $(K_3,H)$.
The Ramsey number $r(K_3,H)$ is one more than the maximum order of a
Ramsey graph for $(K_3,H)$. Note that a maximal triangle-free supergraph
of a Ramsey graph for $(K_3,H)$ of the same order is a Ramsey graph for
$(K_3,H)$ as well.
More details can be found in \cite{Bri96} and \cite{BBH97}. 

The amount of time needed to compute a single Ramsey number $r(K_3,H)$
turns out to depend mainly on the magnitude of $r(K_3,H)$. Even though
this Ramsey number is relatively small for most of the graphs considered here,
the huge number of graphs under investigation makes it impossible to
compute all the Ramsey numbers separately. 
The most time-consuming part is the subgraph testing routine, so, in order to
improve the performance, we tried to reduce the number of subgraph tests. One 
method to do so is to test a group of graphs $H_0, H_1, \dots, H_t$ simultaneously.

These graphs are ordered with respect to subgraph relations, so whenever a graph
$H_i$ is found to be contained in the complement of
a maximal triangle-free graph $F$, all the graphs that
are subgraphs of $H_i$ need not be tested, and -- the other way round -- whenever
a graph $H_j$ is found to be {\bf not} contained in  
the complement of a maximal triangle-free graph $F$, all the graphs that
are supergraphs of $H_j$ need not be tested any more. Some tests showed that
usually it is most efficient to start
testing the minimal elements of the subgraph chains and then proceed to
the larger ones.

This method was used e.g.\ for the graphs $K_9 - iK_2$ for $2 \le i \le 4$.

For very large groups this method is not optimal. The graphs
have to be kept
in the main memory of the computer for quick access, which requires machines
with a lot of memory. Furthermore a lot of useless work is 
done if the subgraph chains are traversed in a direction where no information is gained.
%On the other hand the following problem occurs: Suppose that $H$ is the 
%maximal element of a long subgraph chain 
%of graphs known to have Ramsey number larger than $r-1$
%that shall be tested for Ramsey number $r$.
%If $H$ has Ramsey number $r$, all the subgraphs also have this Ramsey
%number, so all we have to know is that they are subgraphs of $H$.
%Nevertheless for many triangle-free graphs $F$ of order at most $r-1$,
%$H$ might not yet be contained in the complement, so that its subgraphs
%would have been tested on this level (what later turns out to be
%superfluous, but would have been speeding up the computation if $r(K_3,H)>r$).

So for larger lists we optimized the methods already used in \cite{Bri96}, developing a strategy which has the following property: to generate the Ramsey numbers for all connected graphs of order $n$ with Ramsey number at most $r_0$, only the Ramsey numbers of the maximal elements of the subgraph lattice with Ramsey number $r\le r_0$ are actually computed and the time needed for the whole computation is dominated by the time needed to compute these Ramsey numbers. As $n$ grows, the number of maximal elements becomes negligible compared to the
number of graphs altogether. Our approach is as follows:

Assume that we already know the graphs of order $n$ with Ramsey number 
smaller than $r$.
Testing graphs for Ramsey number $r$ we give a (possibly empty) list of
MINGRAPHs, that are graphs
of order at most $n$ which are known to have Ramsey number 
larger than $r$. Furthermore we have a list of MAXGRAPHs, that are the maximal elements
(w.r.t.\ inclusion) of the subgraph lattice with Ramsey number $r$, which is empty in
the beginning,
and, finally, a list of
RAMSEYGRAPHs, containing triangle-free graphs of order $r$ which are (or might be)
Ramsey graphs for $(K_3,H)$ for some graph $H$ of order $n$. This list can be empty
in the beginning, but may also contain graphs which we consider to be candidates for
being Ramsey graphs.

Then the basic structure of the algorithm can be explained as follows: 

\begin{tabbing}
{\bf for} \= $k={n\choose 2}$ {\bf downto} $n-1$ {\bf do}\\
\> {\bf for} \= every connected graph $H$ with $k$ edges in the list {\bf do}\\
\> \> {\bf if} \= $H$ is not contained in any MAXGRAPH {\bf then}\\
\> \> \> {\bf if} \= $H$ is not supergraph of one of the MINGRAPHs {\bf then}\\
\> \> \> \> {\bf if} \= $H$ is contained in the complement of every
RAMSEYGRAPH {\bf then}\\
\> \> \> \> \> {\bf if} \= \mtf{} applied to $H$ finds a Ramsey graph of
order $r$ {\bf then}\\
\> \> \> \> \> \> add this Ramsey graph to the list of RAMSEYGRAPHs; $r(K_3,H)>r$\\
\> \> \> \> \> {\bf else} add $H$ to the list of MAXGRAPHs; $r(K_3,H)\le r$\\
\> \> \> \> {\bf else} $r(K_3,H)>r$\\
\> \> \> {\bf else} $r(K_3,H)>r$\\ 
\> \> {\bf else} $r(K_3,H)\le r$.\\
\end{tabbing}



MAXGRAPHs and RAMSEYGRAPHs are always ordered according to the number of times they
could be used to determine the Ramsey number of a graph, so that the graphs are first
tested against the most promising MAXGRAPHSs and RAMSEYGRAPHs.

Graphs with the same number of edges can be tested in parallel. We
ran this program on a large cluster of workstations in Berlin and Bielefeld 
using the program {\tt autoson}
\cite{McKay96}
to distribute the jobs. Only some small
amount of communication between the processes was necessary in order to
distribute new RAMSEYGRAPHs and MAXGRAPHs. This was done by using a common file system.
The main problem was that each level $k$ had to be worked out completely before the
computations for smaller sized graphs could start. Some tests starting runs on smaller
size graphs before the larger size computations were completed lead to an enormous
amount of computational overlap, so we
did not follow this strategy any further.
The graphs of order $n$ to be tested were generated by the 
program {\tt makeg} \cite{makeg}.

In our computations the average time needed to compute a MAXGRAPH is significantly longer than the average time needed to compute a RAMSEYGRAPH, while the time needed to test a graph against all existing MIN-, MAX-, and RAMSEYGRAPHs is negligible. Since typically more MAXGRAPHs are generated than RAMSEYGRAPHs, the time needed for the whole computation is dominated by the time needed to determine the MAXGRAPHs. So significant improvements in the running time, which are necessary for extending the results, requires improvements in the computation process of a single Ramsey number $r(K_3,H)$.

To compute $r(K_3,H)$ for the graphs of order 9 and 10,
we used the method described above, some preliminary methods on the way from those
described in 
\cite{Bri96}, and some runs testing small groups.

Our data suggest the following conjecture:

\begin{conj}
For every integer $r>10$ there is a connected graph
$H$ with \\ $r(K_3,H)=r$.
\end{conj}

There are no such graphs for $r=1,2,4,8,10$ but we believe that for every
other order such a graph exists. 

\section{The functions $f(K_3,n)$ and $g(K_3,n)$}

As already mentioned in the introduction, the function $f(K_p,n)$ is
the largest integer, such that every connected graph $H$ of order $n$
and size at most $f(K_p,n)$ satisfies $r(K_p,H) = (p-1)(n-1)+1$, and the
function $g(K_p,n)$ is the largest integer, such that there exists a
graph $H$ of order $n$ with $g(K_p,n)$ edges satisfying $r(K_p,H) =
(p-1)(n-1)+1$. Since trees attain the indicated bound, $f(K_p,n)$ and
$g(K_p,n)$ are well defined.

Not much is known about $g(K_p,n)$. Burr et.al.~\cite{BEFRS80} proved that
\[\Omega (n^{p/(p-1)}) \le g(K_p,n) \le \bigO (n^{(p+2)/p} (\log
n)^{1-(\frac{2}{(p^2-p)})})\]
and improved the lower bound by a factor of 
$\sqrt{\log n}$ for $p=3$. 
There is a substantial gap between the upper and the lower bound, and
even a reasonable conjecture for the structure of the graphs 
determining $g(K_p,n)$
seems to be missing. Based on the graphs giving the
lower bound for $p=3$, a possible structure for the graphs determining
$g(K_3,n)$ was proposed by Faudree, Rousseau and Schelp~\cite[Question
2.32]{FRS96}. This structure consists of the disjoint union of complete
graphs and an additional vertex joined to all other vertices. The
values and graphs for $n\le 12$ which we computed are still too small
to judge, but we would rather expect graphs of larger connectivity to
determine $g(K_3,n)$.

The situation is different for $f(K_p,n)$. For $p\ge 4$ it was shown by
Burr et.al.~\cite{BEFRS80}, that $f(K_p,n) = n+o(n)$, while for $p=3$
they proved that $f(K_3,n) > 17n/15$ for $n\ge 4$. The general
belief---expressed in a number of conjectures---was that the growth of
$f(K_3,n)$ is superlinear in $n$. Motivated by the present results,
Brandt~\cite{Bra96} proved that for every constant $c$, almost all
(with probability tending to $1$)
regular graphs $H$ of sufficiently large degree $d$ have Ramsey number
$r(G,H) > c|H|$ for every non-bipartite graph $G$. This implies that
$f(K_3,n)$ is linear in $n$, and in fact it can be shown that $f(K_3,n)
< 12n$ for large $n$. This complements a result of Burr
et.~al.~\cite{BEFRS85}, saying that for every bipartite graph $G$
and constant $\Delta$,
every graph $H$ of sufficiently large order $n$ with bounded maximum
degree $\Delta(H)\le \Delta$ is $G$-good, i.e.\ $r(G,H) = n-1+s(G)$.

It is very likely that the bound $f(K_3,n) <12n$ is fairly weak. For
small values of $n$ for which we computed the exact numbers we have
$f(K_3,n) < 5n/2$, but it seems likely that the exact values for larger
$n$ are somewhat larger. Though it seems difficult to guess a precise
bound for $f(K_3,n)/n$ as $n\to \infty$, it may be simpler to guess the
right Ramsey graphs, i.e.\ the triangle-free graphs $F$ of order $2n-1$
for which there is a connected graph $H$ of order $n$ and size
$f(K_3,n)+1$ which is not contained in the complement of $F$. Possibly
the typical Ramsey graphs are obtained from the lexicographic products
$C_5[\quer{K}_k]$ (obtained from a $5$-cycle by replacing every vertex
by a set of $k$ independent vertices and joining any two sets if the
corresponding vertices were adjacent in the $5$-cycle) by deleting some
vertices to adjust the order. This
is the case for many small $n$ and for large $n$ these graphs suffice
to show that $f(K_3,n)$ has linear growth in $n$. So we pose the
following problem:

\begin{prob} \label{p:sep}
Is it true that for every sufficiently large integer $n$ there is a
graph obtained from $C_5[\quer{K}_k]$, $k=\lceil (2n-1)/5\rceil$, by
deleting $ 5\lceil (2n-1)/5\rceil -2n+1$ vertices, which is a
Ramsey graph for $(K_3,H)$, where $H$ is a
connected graph of order $n$ and size
$f(K_3,n)+1$?
\end{prob}

The strong impact of these graphs would also explain the strange behaviour
of $f(K_3,n)$. Note that $f(K_3,11) = f(K_3,12) = 23$, which was a
surprising fact for us. In fact, we do not even know whether $f(K_3,n)$
is a monotonously increasing function in $n$. If the answer to Problem~\ref{p:sep} was
affirmative, this would probably turn the determination of $f(K_3,n)$
into a separator type problem, essentially asking for the smallest size of a graph
of order $n$ without a $(\frac{1}{3}$--$\frac{2}{3})$-separator of
cardinality at most $2n/5$. A $(\frac{1}{3}$--$\frac{2}{3})$-separator
is a vertex set $X\subseteq V(G)$ such that $V(G)-X$ can be partitioned into
two disjoint sets $A,B$ with $|A|\le |B|\le 2|A|$ with no edge joining a
vertex of $A$ to a vertex of $B$.

Extending the above question even further, for $f(K_p,n)$, $p\ge 4$,
the upper bound on $f(K_p,n)$ suggests that the classical Ramsey graphs
(i.e\ those for $(K_p,K_r)$) might be
among the relevant Ramsey graphs, which have a complex structure in
contrast to the (possibly) simply structured Ramsey graphs for $p=3$.
Now, if $G$ is non-bipartite with clique number $p=2$, what are the
relevant Ramsey graphs determining $f(G,n)$? The smallest example to look at
is $G=C_5$.

\subsection*{Computing explicit values for $f(K_3,n)$ and $g(K_3,n)$}

We computed all values of $f(K_3,n)$ and $g(K_3,n)$ for $n\le
12$. The results are presented in Table 1.

For $n \le 10$ we determined all $K_3$-good graphs, i.e.\ those with Ramsey
number $2n-1$ by the above methods, so the values of $f(K_3,n)$ and $g(K_3,n)$
could be directly read off the lists.
This showed some unexpected properties of the $K_3$-good graphs:
The maximal elements of the subgraph chains are {\bf all} much larger than the
smallest graphs with larger Ramsey number. So we defined $h(K_3,n)$ to be the
smallest number $h$ so that there is a graph $G$ with $n$ vertices and Ramsey number
$2n-1$ that is not contained in a larger graph with this property. Obviously
$f(K_3,n) \le h(K_3,n) \le  g(K_3,n)$, but in all the observed cases $h(K_3,n)$
is much closer to $g(K_3,n)$ than to $f(K_3,n)$.

\begin{prob}
What is the relation between $f(K_3,n), h(K_3,n)$ and $g(K_3,n)$?
Is it true in general that $h(K_3,n)/f(K_3,n) \ge g(K_3,n)/h(K_3,n)$ ?
\end{prob}

Because of the enormous amount of graphs and the quickly increasing time for computing a
single Ramsey number, for $n \ge 11$ we could not determine all graphs with 
Ramsey number $2n-1$. For $n=10$ there were already 334 maximal elements of the subgraph 
chains, 23 of them with $g(K_3,10)$ edges. For $n=11$ we computed 151 maximal elements with
$g(K_3,11)$ edges before we stopped the computations and looked for a faster way.

Another astonishing property of the class of $K_3$-good graphs made it possible
to determine even $f(K_3,12)$: It turned out that in general only very few 
graphs with $g(K_3,n)$ edges are needed to decide that all graphs with $f(K_3,n)$ edges
have Ramsey number $2n-1$. To be precise: All graphs on 10 vertices with $f(K_3,10)$
edges are contained in the first two graphs on 10 vertices with $g(K_3,10)$ edges which we
computed. 
For $n=11$ the first 19 graphs were needed, but taking just graph number 1 and graph
number 19 again gave a list of only two graphs that contained every graph on 11 vertices
with $f(K_3,11)$ edges. 


In general, when testing a graph, \mtf{} was much faster
in finding Ramsey graphs than determining that no Ramsey graph of the given order exists for the testgraph.
So our strategy was as follows: We guessed an upper bound $b_g$ for $g(K_3,n)$ and tested all
graphs with $b_g$ edges for being $K_3$-good. The result was either a number of Ramsey graphs
or a set of graphs which are $K_3$-good or both. In case of $K_3$-good graphs we increased $b_g$
and tested again, in case of no $K_3$-good graphs we decreased $b_g$ and tested the smaller
graphs. As soon as some $K_3$-good graph with $e$ edges is found and all graphs with $e+1$
edges are shown not to be $K_3$-good, we know $g(K_3,n)=e$. Then the set
of maximal $K_3$-good graphs known
so far is taken and tested against a guess $b_f$ for a lower bound of $f(K_3,n)$. In case that
all graphs with $n$ vertices and $b_f$ edges are contained in some of the maximal $K_3$-good graphs, we
increase $b_f$, otherwise we either have to generate more 
of the $K_3$-good graphs of large size
or to test the graphs separately whose
Ramsey number could not be determined in this way. For $n=12$ two $K_3$-good graphs with $g(K_3,12)$
edges and two with $g(K_3,12)-1$ edges were sufficient to contain all graphs on 12 vertices
with 23 edges. For 24 edges one graph was determined separately not to be $K_3$-good, showing
$f(K_3,12)=23$. In all, about $10^9$ Ramsey numbers were determined to compute
$f(K_3,11),f(K_3,12),g(K_3,11)$ and $g(K_3,12)$ -- most of them by showing that the graphs are subgraphs
of another graph formerly shown to be $K_3$-good.

\begin{prob}
Is there always a small set $S$ of $K_3$-good graphs of order
$n$, such that every graph on $n$ vertices with $f(K_3,n)$ edges is contained in an element of $S$ ?
\end{prob}

We do not think that the cardinality of $S$ can be bounded by a constant but it might be bounded by a
moderately growing function in $n$.

\section{Notes and Acknowledgements}

We do not think that the probability for an error in a computer assisted proof is higher than
one in a long proof by hand. But we think that -- whenever possible -- 
a computer assisted proof
should be checked by an independent program. So
although we were very careful in implementing the algorithms and checked our results
against all available data, we think that an independent approach on the calculation
of triangle Ramsey numbers would be an important thing to do.

\medskip

Since the program was run on large clusters of different types of
workstations we could not 
track the amount of CPU used in all cases. So we have no exact values for the
total amount of CPU used.
For example the accumulated CPU time for 
the computation of all Ramsey graphs and all maximal graphs needed to determine the
graphs on 10 vertices with Ramsey number 19 was a bit less than 9 days on a mixed
cluster of sun, sgi and alpha workstations and 133MHz linux PCs. 
The most expensive of the maximal graphs on 
10 vertices with Ramsey number 23 {\bf alone} took almost 22 days 
(distributed over the same cluster). One of the
maximal graphs on 12 vertices with 49 edges and Ramsey number 23 needed 1.6 CPU years
in all, the other one needed about 185 days of CPU.
So the total amount of CPU used is in the range of several CPU-years. 
The program {\tt mtf} or computer readable lists of the graphs in 
this article can be obtained from the authors.

\medskip

We would like to thank our departments for the extensive use of their computers and especially
the group of Prof.~Wachsmuth at
the Technische Fakult\"at in Bielefeld 
for the opportunity to run a lot of jobs on their machines.
Without this support the extensive computations would not have been possible.


\newpage


\begin{center}
\small
\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
\hline
  & $|H|=3$ & $|H|=4$ & $|H|=5$ & $|H|=6$ & $|H|=7$ & $|H|=8$ & $|H|=9$ & $|H|=10$ \\
\hline
$r=5$ & 1 & & & & &  &  & \\
\hline
$r=6$ &1 & & & & &  &  & \\
\hline
$r=7$ &  & 5 & & & &  &  & \\
\hline
$r=8$ &  &   & & & &  &  & \\
\hline
$r=9$ &  & 1 & 18 & & &  &  & \\
\hline
$r=10$ & &   &    & & &  &  & \\
\hline
$r=11$ & &   & 2  & 98 & &  &  & \\
\hline
$r=12$ & &   &    & 6  & &  &  & \\
\hline
$r=13$ & &   &    & 2  & 772 &  &  & \\
\hline
$r=14$ & &   & 1  & 4  &  40 &  &  & \\
\hline
$r=15$ & &   &    &    &     & 9024  &  & \\
\hline
$r=16$ & &   &    &    &  13 & 1440  &  & \\
\hline
$r=17$ & &   &    & 1  &  19 &  498  & 242773  & \\
\hline
$r=18$ & &   &    & 1  &   7 &  119  & 16024  & \\
\hline
$r=19$ & &   &    &    &     &       & 311  & 10 101 711\\
\hline
$r=20$ & &   &    &    &     &      &  & 504\\
\hline
$r=21$ & &   &    &    &   1 &  28    & 1809  & 1 602 240\\
\hline
$r=22$ & &   &    &    &     &       & 22  & 3 155\\
\hline
$r=23$ & &   &    &    &   1 &  6    & 98  & 6 960 \\
\hline
$r=24$ & &   &    &    &     &       &  &   \\
\hline
$r=25$ & &   &    &    &     &  1    & 26  &  ? \\
\hline
$r=26$ & &   &    &    &     &       & 5  &  ? \\
\hline
$r=27$ & &   &    &    &     &       & 3  &  ? \\
\hline
$r=28$ & &   &    &    &     &  1    & 7  &  ? \\
\hline
$r=31$ & &   &    &    &     &       & 1  &  ? \\
\hline
$r=36$ & &   &    &    &     &       & 1  &  ? \\
\hline
\end{tabular}

\medskip

Table 1: Numbers of connected graphs $H$ with triangle Ramsey number $r(K_3,H)= r$.

\end{center} 



\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
  & $f(K_3,n)$ & $g(K_3,n)$ & $h(K_3,n)$ \\
\hline
n=3 & 2 & 2 & 2 \\
\hline
n=4 & 5  & 5 & 5 \\
\hline
n=5 & 7 & 8 & 8 \\
\hline
n=6 & 8 & 12 & 12 \\
\hline
n=7 & 11 & 16 & 15 \\
\hline
n=8 & 11 & 20 & 18\\
\hline
n=9 & 16 & 27 & 24 \\
\hline
n=10 & 18 & 33 & 30 \\
\hline
n=11 & 23 & 41 & \\
\hline
n=12 & 23 & 49 & \\
\hline
\end{tabular}

\medskip

Table 2: Values for $f(K_3,n)$, $g(K_3,n)$ and $h(K_3,n)$.

\end{center} 


\bigskip

\subsection*{\underline{The triangle Ramsey number for connected graphs of order 9}}

{\bf $r(K_3,H)=36$ } if and only if $H=K_9$.

\bigskip

\hrule \bigskip

{\bf $r(K_3,H)=31$ } if and only if $H=K_9-e$.

\bigskip

\hrule \bigskip



{\bf $r(K_3,H)=28$ } if and only if $H^c$ is one of the graphs

\medskip

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\bigskip

\hrule \bigskip


{\bf $r(K_3,H)=27$ } if and only if $H^c$ is one of the graphs

\medskip

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\bigskip

\hrule \bigskip


{\bf $r(K_3,H)=26$ } if and only if $H^c$ is one of the graphs

\medskip

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\bigskip

\hrule \bigskip



{\bf $r(K_3,H)=25$ } if and only if $H^c$ is contained in one of the graphs

\medskip

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\medskip

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\bigskip

\hrule \bigskip




{\bf $r(K_3,H)=23$ } if and only if $H^c$ is contained in one of the graphs

\medskip

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and contains one of the graphs

\medskip

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\bigskip

\hrule \bigskip



{\bf $r(K_3,H)=22$ } if and only if $H^c$ is contained in one of the graphs

\medskip

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and contains one of the graphs

\medskip

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\bigskip

\hrule \bigskip



{\bf $r(K_3,H)=21$ } if and only if $H^c$ is contained in one of the graphs

\medskip

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\begin{picture}(1.4,1.8)
\leavevmode
\epsfxsize=1.4cm
\epsffile{9er_R21_maxcompl.mu_29.eps}
\end{picture}\par
\end{minipage}
\begin{minipage}[t]{2.2cm}
\begin{picture}(1.4,1.8)
\leavevmode
\epsfxsize=1.4cm
\epsffile{9er_R21_maxcompl.mu_30.eps}
\end{picture}\par
\end{minipage}
\begin{minipage}[t]{2.2cm}
\begin{picture}(1.4,1.8)
\leavevmode
\epsfxsize=1.4cm
\epsffile{9er_R21_maxcompl.mu_31.eps}
\end{picture}\par
\end{minipage}
\begin{minipage}[t]{2.2cm}
\begin{picture}(1.4,1.8)
\leavevmode
\epsfxsize=1.4cm
\epsffile{9er_R21_maxcompl.mu_32.eps}
\end{picture}\par
\end{minipage}
\begin{minipage}[t]{2.2cm}
\begin{picture}(1.4,1.8)
\leavevmode
\epsfxsize=1.4cm
\epsffile{9er_R21_maxcompl.mu_33.eps}
\end{picture}\par
\end{minipage}
\begin{minipage}[t]{2.2cm}
\begin{picture}(1.4,1.8)
\leavevmode
\epsfxsize=1.4cm
\epsffile{9er_R21_maxcompl.mu_34.eps}
\end{picture}\par
\end{minipage}
\begin{minipage}[t]{2.2cm}
\begin{picture}(1.4,1.8)
\leavevmode
\epsfxsize=1.4cm
\epsffile{9er_R21_maxcompl.mu_35.eps}
\end{picture}\par
\end{minipage}
\begin{minipage}[t]{2.2cm}
\begin{picture}(1.4,1.8)
\leavevmode
\epsfxsize=1.4cm
\epsffile{9er_R21_maxcompl.mu_36.eps}
\end{picture}\par
\end{minipage}
\begin{minipage}[t]{2.2cm}
\begin{picture}(1.4,1.8)
\leavevmode
\epsfxsize=1.4cm
\epsffile{9er_R21_maxcompl.mu_37.eps}
\end{picture}\par
\end{minipage}
\begin{minipage}[t]{2.2cm}
\begin{picture}(1.4,1.8)
\leavevmode
\epsfxsize=1.4cm
\epsffile{9er_R21_maxcompl.mu_38.eps}
\end{picture}\par
\end{minipage}
\begin{minipage}[t]{2.2cm}
\begin{picture}(1.4,1.8)
\leavevmode
\epsfxsize=1.4cm
\epsffile{9er_R21_maxcompl.mu_39.eps}
\end{picture}\par
\end{minipage}
\begin{minipage}[t]{2.2cm}
\begin{picture}(1.4,1.8)
\leavevmode
\epsfxsize=1.4cm
\epsffile{9er_R21_maxcompl.mu_40.eps}
\end{picture}\par
\end{minipage}
\begin{minipage}[t]{2.2cm}
\begin{picture}(1.4,1.8)
\leavevmode
\epsfxsize=1.4cm
\epsffile{9er_R21_maxcompl.mu_41.eps}
\end{picture}\par
\end{minipage}
\begin{minipage}[t]{2.2cm}
\begin{picture}(1.4,1.8)
\leavevmode
\epsfxsize=1.4cm
\epsffile{9er_R21_maxcompl.mu_42.eps}
\end{picture}\par
\end{minipage}
\begin{minipage}[t]{2.2cm}
\begin{picture}(1.4,1.8)
\leavevmode
\epsfxsize=1.4cm
\epsffile{9er_R21_maxcompl.mu_43.eps}
\end{picture}\par
\end{minipage}
\begin{minipage}[t]{2.2cm}
\begin{picture}(1.4,1.8)
\leavevmode
\epsfxsize=1.4cm
\epsffile{9er_R21_maxcompl.mu_44.eps}
\end{picture}\par
\end{minipage}
\begin{minipage}[t]{2.2cm}
\begin{picture}(1.4,1.8)
\leavevmode
\epsfxsize=1.4cm
\epsffile{9er_R21_maxcompl.mu_45.eps}
\end{picture}\par
\end{minipage}
\begin{minipage}[t]{2.2cm}
\begin{picture}(1.4,1.8)
\leavevmode
\epsfxsize=1.4cm
\epsffile{9er_R21_maxcompl.mu_46.eps}
\end{picture}\par
\end{minipage}
\begin{minipage}[t]{2.2cm}
\begin{picture}(1.4,1.8)
\leavevmode
\epsfxsize=1.4cm
\epsffile{9er_R21_maxcompl.mu_47.eps}
\end{picture}\par
\end{minipage}
\begin{minipage}[t]{2.2cm}
\begin{picture}(1.4,1.8)
\leavevmode
\epsfxsize=1.4cm
\epsffile{9er_R21_maxcompl.mu_48.eps}
\end{picture}\par
\end{minipage}
\begin{minipage}[t]{2.2cm}
\begin{picture}(1.4,1.8)
\leavevmode
\epsfxsize=1.4cm
\epsffile{9er_R21_maxcompl.mu_49.eps}
\end{picture}\par
\end{minipage}
\begin{minipage}[t]{2.2cm}
\begin{picture}(1.4,1.8)
\leavevmode
\epsfxsize=1.4cm
\epsffile{9er_R21_maxcompl.mu_50.eps}
\end{picture}\par
\end{minipage}

%\pagebreak

and contains one of the graphs

\medskip

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\begin{minipage}[t]{2.2cm}
\begin{picture}(1.4,1.8)
\leavevmode
\epsfxsize=1.4cm
\epsffile{9er_R21_mincompl.mu_1.eps}
\end{picture}\par
\end{minipage}
\begin{minipage}[t]{2.2cm}
\begin{picture}(1.4,1.8)
\leavevmode
\epsfxsize=1.4cm
\epsffile{9er_R21_mincompl.mu_2.eps}
\end{picture}\par
\end{minipage}
\begin{minipage}[t]{2.2cm}
\begin{picture}(1.4,1.8)
\leavevmode
\epsfxsize=1.4cm
\epsffile{9er_R21_mincompl.mu_3.eps}
\end{picture}\par
\end{minipage}
\begin{minipage}[t]{2.2cm}
\begin{picture}(1.4,1.8)
\leavevmode
\epsfxsize=1.4cm
\epsffile{9er_R21_mincompl.mu_4.eps}
\end{picture}\par
\end{minipage}
\begin{minipage}[t]{2.2cm}
\begin{picture}(1.4,1.8)
\leavevmode
\epsfxsize=1.4cm
\epsffile{9er_R21_mincompl.mu_5.eps}
\end{picture}\par
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\begin{minipage}[t]{2.2cm}
\begin{picture}(1.4,1.8)
\leavevmode
\epsfxsize=1.4cm
\epsffile{9er_R21_mincompl.mu_6.eps}
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\begin{minipage}[t]{2.2cm}
\begin{picture}(1.4,1.8)
\leavevmode
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\epsffile{9er_R21_mincompl.mu_7.eps}
\end{picture}\par
\end{minipage}
\begin{minipage}[t]{2.2cm}
\begin{picture}(1.4,1.8)
\leavevmode
\epsfxsize=1.4cm
\epsffile{9er_R21_mincompl.mu_8.eps}
\end{picture}\par
\end{minipage}


\bigskip

\hrule \bigskip

{\bf $r(K_3,H)=19$ } if and only if $H^c$ is contained in one of the graphs

\medskip

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\begin{minipage}[t]{2.2cm}
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\epsffile{9er_R19_maxcompl.mu_1.eps}
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\begin{minipage}[t]{2.2cm}
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\epsffile{9er_R19_maxcompl.mu_2.eps}
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\begin{minipage}[t]{2.2cm}
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\epsffile{9er_R19_maxcompl.mu_3.eps}
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\begin{minipage}[t]{2.2cm}
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\epsffile{9er_R19_maxcompl.mu_4.eps}
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\begin{minipage}[t]{2.2cm}
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\epsffile{9er_R19_maxcompl.mu_5.eps}
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\begin{minipage}[t]{2.2cm}
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\epsffile{9er_R19_maxcompl.mu_6.eps}
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\begin{minipage}[t]{2.2cm}
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\epsffile{9er_R19_maxcompl.mu_7.eps}
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\begin{minipage}[t]{2.2cm}
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\epsfxsize=1.4cm
\epsffile{9er_R19_maxcompl.mu_8.eps}
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\begin{minipage}[t]{2.2cm}
\begin{picture}(1.4,1.8)
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\epsfxsize=1.4cm
\epsffile{9er_R19_maxcompl.mu_9.eps}
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\begin{minipage}[t]{2.2cm}
\begin{picture}(1.4,1.8)
\leavevmode
\epsfxsize=1.4cm
\epsffile{9er_R19_maxcompl.mu_10.eps}
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\end{minipage}
\begin{minipage}[t]{2.2cm}
\begin{picture}(1.4,1.8)
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\epsfxsize=1.4cm
\epsffile{9er_R19_maxcompl.mu_11.eps}
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\end{minipage}
\begin{minipage}[t]{2.2cm}
\begin{picture}(1.4,1.8)
\leavevmode
\epsfxsize=1.4cm
\epsffile{9er_R19_maxcompl.mu_12.eps}
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\end{minipage}
\begin{minipage}[t]{2.2cm}
\begin{picture}(1.4,1.8)
\leavevmode
\epsfxsize=1.4cm
\epsffile{9er_R19_maxcompl.mu_13.eps}
\end{picture}\par
\end{minipage}
\begin{minipage}[t]{2.2cm}
\begin{picture}(1.4,1.8)
\leavevmode
\epsfxsize=1.4cm
\epsffile{9er_R19_maxcompl.mu_14.eps}
\end{picture}\par
\end{minipage}
\begin{minipage}[t]{2.2cm}
\begin{picture}(1.4,1.8)
\leavevmode
\epsfxsize=1.4cm
\epsffile{9er_R19_maxcompl.mu_15.eps}
\end{picture}\par
\end{minipage}
\begin{minipage}[t]{2.2cm}
\begin{picture}(1.4,1.8)
\leavevmode
\epsfxsize=1.4cm
\epsffile{9er_R19_maxcompl.mu_16.eps}
\end{picture}\par
\end{minipage}
\begin{minipage}[t]{2.2cm}
\begin{picture}(1.4,1.8)
\leavevmode
\epsfxsize=1.4cm
\epsffile{9er_R19_maxcompl.mu_17.eps}
\end{picture}\par
\end{minipage}
\begin{minipage}[t]{2.2cm}
\begin{picture}(1.4,1.8)
\leavevmode
\epsfxsize=1.4cm
\epsffile{9er_R19_maxcompl.mu_18.eps}
\end{picture}\par
\end{minipage}
\begin{minipage}[t]{2.2cm}
\begin{picture}(1.4,1.8)
\leavevmode
\epsfxsize=1.4cm
\epsffile{9er_R19_maxcompl.mu_19.eps}
\end{picture}\par
\end{minipage}
\begin{minipage}[t]{2.2cm}
\begin{picture}(1.4,1.8)
\leavevmode
\epsfxsize=1.4cm
\epsffile{9er_R19_maxcompl.mu_20.eps}
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\end{minipage}
\begin{minipage}[t]{2.2cm}
\begin{picture}(1.4,1.8)
\leavevmode
\epsfxsize=1.4cm
\epsffile{9er_R19_maxcompl.mu_21.eps}
\end{picture}\par
\end{minipage}
\begin{minipage}[t]{2.2cm}
\begin{picture}(1.4,1.8)
\leavevmode
\epsfxsize=1.4cm
\epsffile{9er_R19_maxcompl.mu_22.eps}
\end{picture}\par
\end{minipage}
\begin{minipage}[t]{2.2cm}
\begin{picture}(1.4,1.8)
\leavevmode
\epsfxsize=1.4cm
\epsffile{9er_R19_maxcompl.mu_23.eps}
\end{picture}\par
\end{minipage}
\begin{minipage}[t]{2.2cm}
\begin{picture}(1.4,1.8)
\leavevmode
\epsfxsize=1.4cm
\epsffile{9er_R19_maxcompl.mu_24.eps}
\end{picture}\par
\end{minipage}
\begin{minipage}[t]{2.2cm}
\begin{picture}(1.4,1.8)
\leavevmode
\epsfxsize=1.4cm
\epsffile{9er_R19_maxcompl.mu_25.eps}
\end{picture}\par
\end{minipage}
\begin{minipage}[t]{2.2cm}
\begin{picture}(1.4,1.8)
\leavevmode
\epsfxsize=1.4cm
\epsffile{9er_R19_maxcompl.mu_26.eps}
\end{picture}\par
\end{minipage}
\begin{minipage}[t]{2.2cm}
\begin{picture}(1.4,1.8)
\leavevmode
\epsfxsize=1.4cm
\epsffile{9er_R19_maxcompl.mu_27.eps}
\end{picture}\par
\end{minipage}
\begin{minipage}[t]{2.2cm}
\begin{picture}(1.4,1.8)
\leavevmode
\epsfxsize=1.4cm
\epsffile{9er_R19_maxcompl.mu_28.eps}
\end{picture}\par
\end{minipage}
\begin{minipage}[t]{2.2cm}
\begin{picture}(1.4,1.8)
\leavevmode
\epsfxsize=1.4cm
\epsffile{9er_R19_maxcompl.mu_29.eps}
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\end{minipage}
\begin{minipage}[t]{2.2cm}
\begin{picture}(1.4,1.8)
\leavevmode
\epsfxsize=1.4cm
\epsffile{9er_R19_maxcompl.mu_30.eps}
\end{picture}\par
\end{minipage}
\begin{minipage}[t]{2.2cm}
\begin{picture}(1.4,1.8)
\leavevmode
\epsfxsize=1.4cm
\epsffile{9er_R19_maxcompl.mu_31.eps}
\end{picture}\par
\end{minipage}
\begin{minipage}[t]{2.2cm}
\begin{picture}(1.4,1.8)
\leavevmode
\epsfxsize=1.4cm
\epsffile{9er_R19_maxcompl.mu_32.eps}
\end{picture}\par
\end{minipage}
\begin{minipage}[t]{2.2cm}
\begin{picture}(1.4,1.8)
\leavevmode
\epsfxsize=1.4cm
\epsffile{9er_R19_maxcompl.mu_33.eps}
\end{picture}\par
\end{minipage}
\begin{minipage}[t]{2.2cm}
\begin{picture}(1.4,1.8)
\leavevmode
\epsfxsize=1.4cm
\epsffile{9er_R19_maxcompl.mu_34.eps}
\end{picture}\par
\end{minipage}
\begin{minipage}[t]{2.2cm}
\begin{picture}(1.4,1.8)
\leavevmode
\epsfxsize=1.4cm
\epsffile{9er_R19_maxcompl.mu_35.eps}
\end{picture}\par
\end{minipage}
\begin{minipage}[t]{2.2cm}
\begin{picture}(1.4,1.8)
\leavevmode
\epsfxsize=1.4cm
\epsffile{9er_R19_maxcompl.mu_36.eps}
\end{picture}\par
\end{minipage}
\begin{minipage}[t]{2.2cm}
\begin{picture}(1.4,1.8)
\leavevmode
\epsfxsize=1.4cm
\epsffile{9er_R19_maxcompl.mu_37.eps}
\end{picture}\par
\end{minipage}
\begin{minipage}[t]{2.2cm}
\begin{picture}(1.4,1.8)
\leavevmode
\epsfxsize=1.4cm
\epsffile{9er_R19_maxcompl.mu_38.eps}
\end{picture}\par
\end{minipage}
\begin{minipage}[t]{2.2cm}
\begin{picture}(1.4,1.8)
\leavevmode
\epsfxsize=1.4cm
\epsffile{9er_R19_maxcompl.mu_39.eps}
\end{picture}\par
\end{minipage}
\begin{minipage}[t]{2.2cm}
\begin{picture}(1.4,1.8)
\leavevmode
\epsfxsize=1.4cm
\epsffile{9er_R19_maxcompl.mu_40.eps}
\end{picture}\par
\end{minipage}
\begin{minipage}[t]{2.2cm}
\begin{picture}(1.4,1.8)
\leavevmode
\epsfxsize=1.4cm
\epsffile{9er_R19_maxcompl.mu_41.eps}
\end{picture}\par
\end{minipage}
\begin{minipage}[t]{2.2cm}
\begin{picture}(1.4,1.8)
\leavevmode
\epsfxsize=1.4cm
\epsffile{9er_R19_maxcompl.mu_42.eps}
\end{picture}\par
\end{minipage}
\begin{minipage}[t]{2.2cm}
\begin{picture}(1.4,1.8)
\leavevmode
\epsfxsize=1.4cm
\epsffile{9er_R19_maxcompl.mu_43.eps}
\end{picture}\par
\end{minipage}
\begin{minipage}[t]{2.2cm}
\begin{picture}(1.4,1.8)
\leavevmode
\epsfxsize=1.4cm
\epsffile{9er_R19_maxcompl.mu_44.eps}
\end{picture}\par
\end{minipage}
\begin{minipage}[t]{2.2cm}
\begin{picture}(1.4,1.8)
\leavevmode
\epsfxsize=1.4cm
\epsffile{9er_R19_maxcompl.mu_45.eps}
\end{picture}\par
\end{minipage}
\begin{minipage}[t]{2.2cm}
\begin{picture}(1.4,1.8)
\leavevmode
\epsfxsize=1.4cm
\epsffile{9er_R19_maxcompl.mu_46.eps}
\end{picture}\par
\end{minipage}


and contains one of the graphs

\medskip

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\begin{picture}(1.4,1.8)
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\epsfxsize=1.4cm
\epsffile{9er_R19_mincompl.mu_1.eps}
\end{picture}\par
\end{minipage}
\begin{minipage}[t]{2.2cm}
\begin{picture}(1.4,1.8)
\leavevmode
\epsfxsize=1.4cm
\epsffile{9er_R19_mincompl.mu_2.eps}
\end{picture}\par
\end{minipage}
\begin{minipage}[t]{2.2cm}
\begin{picture}(1.4,1.8)
\leavevmode
\epsfxsize=1.4cm
\epsffile{9er_R19_mincompl.mu_3.eps}
\end{picture}\par
\end{minipage}
\begin{minipage}[t]{2.2cm}
\begin{picture}(1.4,1.8)
\leavevmode
\epsfxsize=1.4cm
\epsffile{9er_R19_mincompl.mu_4.eps}
\end{picture}\par
\end{minipage}
\begin{minipage}[t]{2.2cm}
\begin{picture}(1.4,1.8)
\leavevmode
\epsfxsize=1.4cm
\epsffile{9er_R19_mincompl.mu_5.eps}
\end{picture}\par
\end{minipage}
\begin{minipage}[t]{2.2cm}
\begin{picture}(1.4,1.8)
\leavevmode
\epsfxsize=1.4cm
\epsffile{9er_R19_mincompl.mu_6.eps}
\end{picture}\par
\end{minipage}
\begin{minipage}[t]{2.2cm}
\begin{picture}(1.4,1.8)
\leavevmode
\epsfxsize=1.4cm
\epsffile{9er_R19_mincompl.mu_7.eps}
\end{picture}\par
\end{minipage}
\begin{minipage}[t]{2.2cm}
\begin{picture}(1.4,1.8)
\leavevmode
\epsfxsize=1.4cm
\epsffile{9er_R19_mincompl.mu_8.eps}
\end{picture}\par
\end{minipage}
\begin{minipage}[t]{2.2cm}
\begin{picture}(1.4,1.8)
\leavevmode
\epsfxsize=1.4cm
\epsffile{9er_R19_mincompl.mu_9.eps}
\end{picture}\par
\end{minipage}
\begin{minipage}[t]{2.2cm}
\begin{picture}(1.4,1.8)
\leavevmode
\epsfxsize=1.4cm
\epsffile{9er_R19_mincompl.mu_10.eps}
\end{picture}\par
\end{minipage}
\begin{minipage}[t]{2.2cm}
\begin{picture}(1.4,1.8)
\leavevmode
\epsfxsize=1.4cm
\epsffile{9er_R19_mincompl.mu_11.eps}
\end{picture}\par
\end{minipage}
\begin{minipage}[t]{2.2cm}
\begin{picture}(1.4,1.8)
\leavevmode
\epsfxsize=1.4cm
\epsffile{9er_R19_mincompl.mu_12.eps}
\end{picture}\par
\end{minipage}
\begin{minipage}[t]{2.2cm}
\begin{picture}(1.4,1.8)
\leavevmode
\epsfxsize=1.4cm
\epsffile{9er_R19_mincompl.mu_13.eps}
\end{picture}\par
\end{minipage}
\begin{minipage}[t]{2.2cm}
\begin{picture}(1.4,1.8)
\leavevmode
\epsfxsize=1.4cm
\epsffile{9er_R19_mincompl.mu_14.eps}
\end{picture}\par
\end{minipage}
\begin{minipage}[t]{2.2cm}
\begin{picture}(1.4,1.8)
\leavevmode
\epsfxsize=1.4cm
\epsffile{9er_R19_mincompl.mu_15.eps}
\end{picture}\par
\end{minipage}
\begin{minipage}[t]{2.2cm}
\begin{picture}(1.4,1.8)
\leavevmode
\epsfxsize=1.4cm
\epsffile{9er_R19_mincompl.mu_16.eps}
\end{picture}\par
\end{minipage}
\begin{minipage}[t]{2.2cm}
\begin{picture}(1.4,1.8)
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\bigskip

\hrule \bigskip


{\bf $r(K_3,H)=18$ } if and only if $H^c$ is contained in one of the graphs

\medskip

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%\pagebreak
 
and contains one of the graphs

\medskip

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\bigskip

\hrule \bigskip


{\bf $r(K_3,H)=17$ } if and only if $H^c$ contains one of the graphs

\nopagebreak

\medskip

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\bigskip

\subsection*{\underline{Extremal graphs for $f(K_3,n)$ and $g(K_3,n)$}}




The complements of the graphs $H$ on 10 vertices 
with $r(K_3,H)=19$ and \\ $|E(H)|=g(K_3,10)$.

\medskip



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\bigskip

\hrule \bigskip

The graphs $H$ on 10 vertices 
with $r(K_3,H)>19$ and \\ $|E(H)|=19=f(K_3,10)+1$.

\medskip

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\bigskip

\hrule \bigskip

Two  complements of graphs $H$ on 10 vertices 
with $r(K_3,H)=19$ and \\ $|E(H)|=33=g(K_3,10)$ 
containing all graphs on 10 vertices
with up to $f(K_3,10)=18$ edges.

\medskip

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\bigskip



\hrule \bigskip

The unique graph $H$ on 11 vertices 
with $r(K_3,H)>21$ and \\ $|E(H)|=24=f(K_3,11)+1$.
\nopagebreak

\medskip
\nopagebreak
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\end{picture}\par
\end{minipage}

\bigskip

\hrule \bigskip

Two  complements of graphs $H$ on 11 vertices 
with $r(K_3,H)=21$ and \\ $|E(H)|=41=g(K_3,11)$
containing all graphs on 11 vertices
with up to $f(K_3,11)=23$ edges.



\medskip

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\end{picture}\par
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\bigskip

%\pagebreak

\hrule \bigskip

An example graph $H$ on 12 vertices 
with $r(K_3,H)>23$ and \\ $|E(H)|=24=f(K_3,12)+1$.

\medskip

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\end{picture}\par
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\bigskip

\hrule \bigskip

Two  complements of graphs $H$ on 12 vertices 
with $r(K_3,H)=23$ and $|E(H)|=49=g(K_3,12)$ and two 
 complements of graphs $H$ 
with $r(K_3,H)=23$ and $|E(H)|=48=g(K_3,12)-1$
containing all graphs on 12 vertices
with up to $f(K_3,12)=23$ edges.

\medskip

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\end{document}