  A typical theme for many well-known decomposition problems is to show that
some obvious necessary conditions for decomposing a graph $G$ into copies $H_1,
\ldots, H_m$ are also sufficient. One such problem was posed in 1987, by Alavi,
Boals, Chartrand, Erd\H{o}s, and Oellerman. They conjectured that the edges of
every graph with $\binom{m+1}2$ edges can be decomposed into subgraphs $H_1,
\dots, H_m$ such that each $H_i$ has $i$ edges and is isomorphic to a subgraph
of $H_{i+1}$. In this paper we prove this conjecture for sufficiently large
$m$.
