Lagrangians of Hypergraphs

Abstract

The Lagrangian of a hypergraph is a function that in a sense seems to measure how ‘tightly packed’ a subset of the hypergraph one can find. Lagrangians were first used by Motzkin and Straus to obtain a new proof of a classic theorem of Turán, and subsequently found a number of very valuable applications in Extremal Hypergraph Theory; one remarkable result they yield is the disproof of a famous "jumping conjecture" of Erdos, which we reprove entirely; we will also introduce a very recent method based on Razborov's flag algebras to show that, though the jumping conjecture is false in general, hypergraphs "do jump" in some cases