Asymptotics of \(t_3(n, 5)\)

A hypergraph \( H\) consists of a vertex set \( V\) together with a family \( E\) of subsets of \( V\), which are called the edges of \( H\). A \( r\)-uniform hypergraph, or \( r\)-graph, for short, is a hypergraph whose edge set consists of \( r\)-subsets of \( V\). A graph is just a special case of an \( r\)-graph with \( r=2\).

For a finite family \( \mathcal F\) of \( r\)-graphs, one can define the Turán number \( t_r(n,{\mathcal F})\) to be the largest integer \( t\) such that there is an \( r\)-graph on \( n\) vertices with \( t\) edges which does not contain any \( F \in {\mathcal F}\) as a subgraph. We let \( K_k^{(r)}\) denote a complete \( r\)-graph on \( k\) vertices and we write \( t_r(n,K_k^{(r)})=t_r(n,k)\).

Conjecture [1]

\[ \lim_{n \rightarrow \infty} \frac{t_3(n,5)}{\binom{n}{3}} = \frac{1}{ 4}. \]

Bibliography
1 P. Turán, On an extremal problem in graph theory (in Hungarian), Mat. Fiz. Lapok. 48 (1941), 436-452. English translation in Collected Papers of Paul Turán (P. Erdös, ed.), 231-240, Akadémiai Kiadó, Budapest, 1990.