One of the earliest results in Combinatorics is Mantel's theorem from 1907 that the largest triangle-free graph on a given vertex set is complete bipartite. However, a seemingly similar question posed by Turán in 1941 is still open: what is the largest 3-uniform hypergraph on a given vertex set with no tetrahedron? This question can be considered a test case for the general hypergraph Turán problem, where given an r-uniform hypergraph F, we want to determine the maximum number of edges in an r-uniform hypergraph on n vertices that does not contain a copy of F. To date there are very few results on this problem, even asymptotically. However, recent years have seen a revitalisation of this field, via significant developments in the available methods, notably the use of stability (approximate structure) and flag algebras. This article surveys the known results and methods, and discusses some open problems.
Research supported in part by ERC grant 239696 and EPSRC grant EP/G056730/1. Thanks to Dan Hefetz, Dhruv Mubayi, Richard Mycroft and Oleg Pikhurko for helpful comments and corrections.
The Turán number ex(n, F) is the maximum number of edges in an F-free r-graph on n vertices. It is a long-standing open problem in Extremal Combinatorics to develop some understanding of these numbers for general r-graphs F. Ideally, one would like to compute them exactly, but even asymptotic results are currently only known in certain cases. For ordinary graphs (r = 2) the picture is fairly complete.
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