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 $r=2$.

Ramsey-Turán problems for graphs can be naturally generalized to hypergraphs [1]. For an $r$-graph $H$ and an integer $k$, we denote by $rt_r(n,H;k)$ the maximum number of edges in an $r$-graph $G$ on $n$ vertices when $G$ contains no independent set of size $k$, and $G$ does not contain a $H$ as a subgraph. The problems of estimating $rt_r(n,H;k)$ for $r$-graphs $H$, $r \geq 3$, are considerably harder than the case for graphs. Known results on this problem are contained in [1] and [2] Numerous problems on Ramsey-Turán type problems are open (e.g., see [1]). Erds and Sós [1] showed that there is an $\alpha < 1$ such that $rt_3(n,K_4^{(3)}; n^{\alpha}) \geq c_{\alpha} n^3$ for a constant $c_{\alpha}$. They asked:

See also this conjecture for the corresponding upper bound.

Bibliography

1

P. Erds and V. T. Sós. On Ramsey-Turán type theorems for hypergraphs. Combinatorica 2 (1982), 289-295.

2

P. Frankl and V. Rödl. Some Ramsey-Turán type results for hypergraphs. Combinatorica 8 (1988), 323-332.