43 research outputs found
The maximal length of a gap between r-graph Tur\'an densities
The Tur\'an density of a family of -graphs is the
limit as of the maximum edge density of an -free -graph
on vertices. Erdos [Israel J. Math 2 (1964) 183--190] proved that no
Tur\'an density can lie in the open interval . Here we show that
any other open subinterval of avoiding Tur\'an densities has strictly
smaller length. In particular, this implies a conjecture of Grosu [E-print
arXiv:1403.4653v1, 2014].Comment: 7 page
Hypergraphs do jump
We say that is a jump for an integer if there
exists such that for all and all any
-graph with vertices and density at least
contains a subgraph on vertices of density at least
. The Erd\H os--Stone--Simonovits theorem implies that for
every is a jump. Erd\H os showed that for all ,
every is a jump. Moreover he made his famous "jumping
constant conjecture" that for all , every is a
jump. Frankl and R\"odl disproved this conjecture by giving a sequence of
values of non-jumps for all . We use Razborov's flag algebra method to
show that jumps exist for in the interval . These are the first
examples of jumps for any in the interval . To be precise
we show that for every is a jump. We also
give an improved upper bound for the Tur\'an density of
: . This in turn implies that for
every is a jump.Comment: 11 pages, 1 figure, 42 page appendix of C++ code. Revised version
including new Corollary 2.3 thanks to an observation of Dhruv Mubay
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