Abstract
Much of extremal graph theory has concentrated either on finding very small subgraphs of a large graph (such as Turán's theorem [Turán, P., On an extremal problem in graph theory, Matematiko Fizicki Lapok 48 (1941), 436-452]) or on finding spanning subgraphs (such as Dirac's theorem [Dirac, G.A., Some theorems on abstract graphs, Proc. London Math.
Soc. s3-2 (1952), 69-81] or more recently work of Komlós, Sárközy and Szemerédi towards a proof of the Pósa-Seymour conjecture). Only a few results give conditions to obtain some intermediate-sized subgraph.
We contend that this neglect is unjustified. To support our contention we focus on the illustrative case of minimum degree conditions which guarantee squared-cycles of various lengths, but also offer results, conjectures and comments on other powers of paths and cycles, generalisations thereof, and hypergraph variants.