Ramsey numbers for triangles versus almost-complete graphs

We show that, in any coloring of the edges of K_38 with two colors, there exists a triangle in the first color or a monochromatic K_10-e (K_10 with one edge removed) in the second color, and hence we obtain a bound on the corresponding Ramsey number, R(K_3, K_10-e) \u3c= 38. the new lower bound of 37 for this number is established by a coloring of K_36 avoiding triangles in the first color and K_10-e in the second color. This improves by one the vest previously known lower and upper bounds. we also give the bounds for the next Ramsey number of this type, 42 \u3c= R(K_3, K_11-e) \u3c= 47

Empty Rectangles and Graph Dimension

We consider rectangle graphs whose edges are defined by pairs of points in diagonally opposite corners of empty axis-aligned rectangles. The maximum number of edges of such a graph on $n$ points is shown to be 1/4 n^2 +n -2. This number also has other interpretations: * It is the maximum number of edges of a graph of dimension \bbetween{3}{4}, i.e., of a graph with a realizer of the form \pi_1,\pi_2,\ol{\pi_1},\ol{\pi_2}. * It is the number of 1-faces in a special Scarf complex. The last of these interpretations allows to deduce the maximum number of empty axis-aligned rectangles spanned by 4-element subsets of a set of $n$ points. Moreover, it follows that the extremal point sets for the two problems coincide. We investigate the maximum number of of edges of a graph of dimension $\between{3}{4}$, i.e., of a graph with a realizer of the form \pi_1,\pi_2,\pi_3,\ol{\pi_3}. This maximum is shown to be $1/4 n^2 + O(n)$. Box graphs are defined as the 3-dimensional analog of rectangle graphs. The maximum number of edges of such a graph on $n$ points is shown to be $7/16 n^2 + o(n^2)$