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A sequence of nonnegative integers \pi =(d_1,d_2,...,d_n) is graphic if thereis a (simple) graph G with degree sequence \pi. In this case, G is said torealize or be a realization of \pi. Degree sequence results in the literaturegenerally fall into two classes: forcible problems, in which all realizationsof a graphic sequence must have a given property, and potential problems, inwhich 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 issome 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 twicethe number of edges in any realization of \pi, the Erd\H{o}s-Jacobson-Lehelproblem can be viewed as a potential degree sequence relaxation of the(forcible) Tur\'{a}n problem, wherein one wishes to determine the maximumnumber 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 ofspecific classes of graphs (including cliques, cycles, complete bigraphs andothers), very little is known about the parameter for arbitrary H. In thispaper, we determine \sigma(H,n) asymptotically for all H, thereby providing anErd\H{o}s-Stone-Simonovits-type theorem for the Erd\H{o}s-Jacobson-Lehelproblem.

On the Sum Neccesary to Ensure that a Degree Sequence is Potentially H-Graphic