  A sequence of nonnegative integers \pi =(d_1,d_2,...,d_n) is graphic if there
is a (simple) graph G with degree sequence \pi. In this case, G is said to
realize or be a realization of \pi. Degree sequence results in the literature
generally fall into two classes: forcible problems, in which all realizations
of a graphic sequence must have a given property, and potential problems, in
which at least one realization of \pi must have the given property.
  Given a graph H, a graphic sequence \pi is potentially H-graphic if there is
some realization of \pi that contains H as a subgraph. In 1991, Erd\H{o}s,
Jacobson and Lehel posed the following question: Determine the minimum integer
\sigma(H,n) such that every n-term graphic sequence with sum at least
\sigma(H,n) is potentially H-graphic. As the sum of the terms of \pi is twice
the number of edges in any realization of \pi, the Erd\H{o}s-Jacobson-Lehel
problem can be viewed as a potential degree sequence relaxation of the
(forcible) Tur\'{a}n problem, wherein one wishes to determine the maximum
number of edges in a graph that contains no copy of H.
  While the exact value of \sigma(H,n) has been determined for a number of
specific classes of graphs (including cliques, cycles, complete bigraphs and
others), very little is known about the parameter for arbitrary H. In this
paper, we determine \sigma(H,n) asymptotically for all H, thereby providing an
Erd\H{o}s-Stone-Simonovits-type theorem for the Erd\H{o}s-Jacobson-Lehel
problem.
