Families of Trees Decompose the Random Graph in Any Arbitrary Way

Let = {, …, } be a family of graphs. A graph with edges is called if for linear combination of the form () + … + () = where each is a nonnegative integer, there is a coloring of the edges of with + … + colors such that exactly color classes induce each a copy of ; for = 1, …, . We prove that if is any fixed family of trees then log is a sharp threshold function for the property that the random graph () is totally -decomposable. In particular, if is a tree, then log is a sharp threshold function for the property that () contains ⌊()/()⌋ edge-disjoint copies of