On the sum necessary to ensure that a degree sequence is potentially H-graphic

Given a graph H, a graphic sequence π is potentially H-graphic if there is some realization of π that contains H as a subgraph. In 1991, Erdős, Jacobson and Lehel posed the following question: Determine the minimum even integer σ(H,n) such that every n-term graphic sequence with sum at least σ(H,n) is potentially H-graphic. This problem can be viewed as a “potential” degree sequence relaxation of the (forcible) Turán problems. While the exact value of σ(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 σ(H,n) asymptotically for all H, thereby providing an Erdős-Stone-Simonovits-type theorem for the Erdős-Jacobson-Lehel problem.