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A graph $G$ is called a $(3,j;n)$-minimal Ramsey graph if it has the leastamount of edges, $e(3,j;n)$, given that $G$ is triangle-free, the independencenumber $\alpha(G) < j$ and that $G$ has $n$ vertices. Triangle-free graphs $G$with $\alpha(G) < j$ and where $e(G) - e(3,j;n)$ is small are said to be almostminimal Ramsey graphs.  We look at a construction of some almost minimal Ramsey graphs, called$H_{13}$-patterned graphs. We make computer calculations of the number ofalmost minimal Ramsey triangle-free graphs that are $H_{13}$-patterned. Theresults of these calculations indicate that many of these graphs are in fact$H_{13}$-patterned. In particular, all but one of the connected$(3,j;n)$-minimal Ramsey graphs for $j \leq 9$ are indeed $H_{13}$-patterned.

A computerised classification of some almost minimal triangle-free Ramsey graphs