A hypergraph consists of a vertex set together with a family of subsets of , which are called the edges of . A -uniform hypergraph, or -graph, for short, is a hypergraph whose edge set consists of -subsets of . A graph is just a special case of an -graph with .
Ramsey-Turán problems for graphs can be naturally generalized to hypergraphs [1]. For an -graph and an integer , we denote by the maximum number of edges in an -graph on vertices when contains no independent set of size , and does not contain a as a subgraph. The problems of estimating for -graphs , , 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]). Here we mention the following problem for -graphs:
See also this conjecture for the corresponding lower bound.