39 research outputs found
On hypergraph Lagrangians
It is conjectured by Frankl and F\"uredi that the -uniform hypergraph with
edges formed by taking the first sets in the colex ordering of
has the largest Lagrangian of all -uniform hypergraphs
with edges in \cite{FF}. Motzkin and Straus' theorem confirms this
conjecture when . For , it is shown by Talbot in \cite{T} that this
conjecture is true when is in certain ranges. In this paper, we explore the
connection between the clique number and Lagrangians for -uniform
hypergraphs. As an implication of this connection, we prove that the
-uniform hypergraph with edges formed by taking the first sets in
the colex ordering of has the largest Lagrangian of all
-uniform graphs with vertices and edges satisfying for
Comment: 10 pages. arXiv admin note: substantial text overlap with
arXiv:1312.7529, arXiv:1211.7057, arXiv:1211.6508, arXiv:1311.140
Lagrangians of Hypergraphs
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