Abstract :
In this paper we investigate edge partition problems of the countable triangle free homogeneous graph. As consequences of the main result, we obtain the following theorems. For every coloring of the edges of the countable triangle free homogeneous graph U with finitely many colors there exists a copy of U in U whose edges are colored with at most two of the colors. The countable triangle free homogeneous graph U is weakly edge indivisible, that is, for every coloring of the edges of U with two colors, say red and blue, the following holds: If there is a finite triangle free graph G so that every copy of G in U contains a red edge, then there is a copy of U in U which has red edges only.
