GENERATING NON-JUMPING NUMBERS OF HYPERGRAPHS

The concept of jump concerns the distribution of $Tur{\acute{a}}n$ densities. A number ${\alpha}\;{\in}\;[0,1)$ is a jump for r if there exists a constant c > 0 such that if the $Tur{\acute{a}}n$ density of a family $\mathfrak{F}$ of r-uniform graphs is greater than ${\alpha}$, then the $Tur{\acute{a}}n$ density of $\mathfrak{F}$ 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 $r_1$, $r_2{\geq}2$ 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 $Tur{\acute{a}}n$ density of the extension of the (r - 3)-fold enlargement of a 3-uniform matching.