We study the number of edges, e(G), in triangle-free graphs with a prescribed number of vertices, n(G), independence number, alpha(G), and number of cycles of length 4, N(C-4; G). In particular we show that 3e(G) - 17n(G) + 35 alpha(G) + N(C-4; G) >= 0 for all triangle-free graphs G. We also characterise the graphs that satisfy this inequality with equality. As a consequence we improve the previously best known lower bounds on the independence ratio i(G) = alpha(G)/n(G) for graphs of average degree at most 4 and girth at least 5, 6 or 7.

A graph G is called a (3,j;n)-minimal Ramsey graph if it has the least amount of edges, e(3,j;n), given that G is triangle-free, the independence number α(G)<j and that G has n vertices. Triangle-free graphs G with α(G)<j and where e(G)−e(3,j;n) is small are said to be almost minimal Ramsey graphs. We look at a construction of some almost minimal Ramsey graphs, called H₁₃-patterned graphs. We make computer calculations of the number of almost minimal Ramsey triangle-free graphs that are H₁₃-patterned. The results of these calculations indicate that many of these graphs are in fact H₁₃-patterned. In particular, all but one of the connected (3,j;n)-minimal Ramsey graphs for j≤9 are indeed H₁₃-patterned.