A. 504. Prove that for arbitrary integers 0<r<k<t there exists a positive integer N(r,k,t) which satisfies the following property: whenever G is an r-uniform hypergraph with at least N(r,k,t) vertices such that there is at least one hyperedge on any k vertices, then G contains a complete subgraph with t vertices. (A hypergraph is a graph whose edges are arbitrary subsets of the vertices. The graph is called r-uniform if all edges contain exactly r vertices. An r-uniform hypergraph is complete if any r of its vertices form an edge.)
Deadline expired on 12 April 2010.