Generating non-jumping numbers of hypergraphs
Bull. Korean Math. Soc. 2019 Vol. 56, No. 4, 1027-1039
https://doi.org/10.4134/BKMS.b180808
Published online July 31, 2019
Shaoqiang Liu, Yuejian Peng
Hunan University; Hunan University
Abstract : The concept of jump concerns the distribution of Tur\'an densities. A number $\alpha\in[0,1)$ is a \emph{jump} for $r$ if there exists a constant $c>0$ such that if the Tur\'an density of a family $\mathscr{F}$ of $r$-uniform graphs is greater than $\alpha$, then the Tur\'an density of $\mathscr{F}$ is at least $\alpha+c$. To determine whether a number is a jump or non-jump has been a challenging problem in extremal hypergraph theory. In this paper, we give a way to generate non-jumps for hypergraphs. We show that if $\alpha,\beta$ are non-jumps for $r_1,r_2\geq2$ respectively, then $\frac{\alpha\beta(r_1+r_2)!r_1^{r_1} r_2^{r_2}}{r_1!r_2!(r_1+r_2)^{r_1+r_2}}$ is a non-jump for $r_1+r_2$. We also apply the Lagrangian method to determine the Tur\'an density of the extension of the $(r-3)$-fold enlargement of a $3$-uniform matching.
Keywords : Tur\'an density, hypergraph, hypergraph Lagrangian
MSC numbers : 05C65, 05D05
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