Universality, Tolerance, Chaos and Order

What is the minimum possible number of edges in a graph that contains a copy of every graph on n vertices with maximum degree a most k ? This question, as well as several related variants, received a considerable amount of attention during the last decade. In this short survey we describe the known results focusing on the main ideas in the proofs, discuss the remaining open problems, and mention a recent application in the investigation of the complexity of subgraph containment problems.

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