Ramsey numbers for triangles versus almost-complete graphs

We show that, in any coloring of the edges of K38 with two colors, there exists a triangle in the first color or a monochromatic K 10-e (K10 with one edge removed) in the second color, and hence we obtain a bound on the corresponding Ramsey number, R(K 3,K10-e) ≤ 38. The new lower bound of 37 for this number is established by a coloring of K36 avoiding triangles in the first color and K10-e in the second color. This improves by one the best previously known lower and upper bounds. We also give the bounds for the next Ramsey number of this type, 42 ≤ R(K3, K11-e) ≤ 47