Random triangles in random graphs

Triangles in random graphs

Triangles in Random Graphs

OF THE DISSERTATION Triangles in Random Graphs by Robert DeMarco Dissertation Director: Jeff Kahn We prove four separate results. These results will appear or have appeared in various papers (see [10], [11], [12], [13]). For a gentler introduction to these results, the reader is directed to the first chapter of this thesis. Let G = Gn,p. With ξk = ξ n,p k the number of copies of Kk in G, p ≥ n−...

Online Ramsey games for triangles in random graphs

In the online F -avoidance edge-coloring game with r colors, a graph on n vertices is generated by at each stage randomly adding a new edge. The player must color each new edge as it appears; his goal is to avoid a monochromatic copy of F . Let N0(F, r, n) be the threshold function for the number of edges that the player is asymptotically almost surely able to paint before he loses. Even when F...

Random Triangles II

Let S denote the unit sphere in Euclidean 3-space. A spherical triangle T is a region enclosed by three great circles on S; a great circle is a circle whose center is at the origin. The sides of T are arcs of great circles and have length a, b, c. Each of these is ≤ π. The angle α opposite side a is the dihedral angle between the two planes passing through the origin and determined by arcs b, c...

Random Triangles III

Let Ω be a compact convex set in Euclidean n-space with nonempty interior. Random triangles are defined here by selecting three independent uniformly distributed points in Ω to be vertices. Generating such points for (n,Ω) = (2,unit square) or (n,Ω) = (3,unit cube) is straightforward. For (n,Ω) = (2,unit disk) or (n,Ω) = (3,unit ball), we use the following result [1]. Let X1, X2, X3, Y1, Y2, Y3...