SEMINARS
Cover Turan number of Berge hypergraphs

2019-07-22　15:30 — 17:00

639

Linyuan LU

University of South Carolina , USA

For a fixed set of positive integers $R$, we say $\cal H$ is an $R$-uniform hypergraph, or $R$-graph, if the cardinality of each edge belongs to $R$. For a graph $G=(V,E)$, a hypergraph $\mathcal{H}$ is called a \textit{Berge}-$G$, denoted by $BG$, if there is

an injection $i\colon V(G)\to V(\mathcal{H})$ and

a bijection $f\colon E(G) \to E(\mathcal{H})$ such that for all $e=uv \in E(G)$, we have $\{i(u), i(v)\} \subseteq f(e)$.

%if there exists a bijection $f: E(G) \to E(\mathcal{H})$ such that for every $e \in E(G)$, $e \subseteq f(e)$.

In this paper, we define a variant of Tur\'an number in hypergraphs, namely the \emph{cover Tur\'an number}, denoted as $\hat{ex}_R(n, G)$, as the maximum number of edges in the shadow graph of a Berge-$G$ free $R$-graph on $n$ vertices. We show a general upper bound on the cover Tur\'an number of graphs and determine the cover Tur\'an density of all graphs when the uniformity of the host hypergraph equals to $3$.

(Joint work with Zhiyu Wang)