Turán H-densities for 3-graphs

Details Turán H-densities for 3-graphs Turán H-densities for 3-graphs

2012-01-20

arxiv.org/abs/1201.4326v1

Mathematics

Combinatorics

22 pages, 3 figures

Scientific paper

Given an $r$-graph $H$ on $h$ vertices, and a family $\mathcal{F}$ of forbidden subgraphs, we define $\ex_{H}(n, \mathcal{F})$ to be the maximum number of induced copies of $H$ in an $\mathcal{F}$-free $r$-graph on $n$ vertices. Then the \emph{Tur\'an $H$-density} of $\mathcal{F}$ is the limit \[\pi_{H}(\mathcal{F})= \lim_{n\rightarrow \infty}\ex_{H}(n, \mathcal{F})/\binom{n}{h}. \] This generalises the notions of \emph{Tur\'an density} (when $H$ is an $r$-edge), and \emph{inducibility} (when $\mathcal{F}$ is empty). Although problems of this kind have received some attention, very few results are known. We use Razborov's semi-definite method to investigate Tur\'an $H$-densities for $3$-graphs. In particular, we show that \[\pi_{K_4^-}(K_4) = 16/27,\] with Tur\'an's construction being optimal. We prove a result in a similar flavour for $K_5$ and make a general conjecture on the value of $\pi_{K_t^-}(K_t)$. We also establish that \[\pi_{4.2}(\emptyset)=3/4,\] where $4.2$ denotes the $3$-graph on $4$ vertices with exactly $2$ edges. The lower bound in this case comes from a random geometric construction strikingly different from previous known extremal examples in $3$-graph theory. We give a number of other results and conjectures for $3$-graphs, and in addition consider the inducibility of certain directed graphs. Let $\vec{S}_k$ be the \emph{out-star} on $k$ vertices; i.e{.} the star on $k$ vertices with all $k-1$ edges oriented away from the centre. We show that \[\pi_{\vec{S}_3}(\emptyset)=2\sqrt3-3,\] with an iterated blow-up construction being extremal. This is related to a conjecture of Mubayi and R\"odl on the Tur\'an density of the 3-graph $C_5$. We also determine $\pi_{\vec{S}_k}(\emptyset)$ when $k=4$, and conjecture its value for general $k$.

Affiliated with

Also associated with

No associations

Rating

Turán H-densities for 3-graphs does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Turán H-densities for 3-graphs, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Turán H-densities for 3-graphs will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-321136