Independence densities of hypergraphs

We consider the number of independent sets in hypergraphs, which allows us to define the independence density of countable hypergraphs. Hypergraph independence densities include a broad family of densities over graphs and relational structures, such as $F$-free densities of graphs for a given graph $F.$ In the case of $k$-uniform hypergraphs, we prove that the independence density is always rational. In the case of finite but unbounded hyperedges, we show that the independence density can be any real number in $[0,1].$ Finally, we extend the notion of independence density via independence polynomials

Strong Jumps and Lagrangians of Non-Uniform Hypergraphs

The hypergraph jump problem and the study of Lagrangians of uniform hypergraphs are two classical areas of study in the extremal graph theory. In this paper, we refine the concept of jumps to strong jumps and consider the analogous problems over non-uniform hypergraphs. Strong jumps have rich topological and algebraic structures. The non-strong-jump values are precisely the densities of the hereditary properties, which include the Tur\'an densities of families of hypergraphs as special cases. Our method uses a generalized Lagrangian for non-uniform hypergraphs. We also classify all strong jump values for $\{1,2\}$-hypergraphs.Comment: 19 page

Hypergraph Tur\'{a}n densities can have arbitrarily large algebraic degree

Grosu [On the algebraic and topological structure of the set of Tur\'{a}n densities. \emph{J. Combin. Theory Ser. B} \textbf{118} (2016) 137--185] asked if there exist an integer $r\ge 3$ and a finite family of $r$-graphs whose Tur\'{a}n density, as a real number, has (algebraic) degree greater than~$r-1$. In this note we show that, for all integers $r\ge 3$ and $d$, there exists a finite family of $r$-graphs whose Tur\'{a}n density has degree at least~$d$, thus answering Grosu's question in a strong form.Comment: 9 pages, 1 figure, short note. Comments are welcome