Improvements of the Frankl–Rödl theorem on the number of edges of a hypergraph with forbidden intersections, and their consequences in the problem of finding the chromatic number of a space with forbidden equilateral triangle

We survey the results (both old and new) related to the classical Frankl–Rödl theorem on the upper bound for the product of cardinalities of edge sets of two hypergraphs satisfying the condition that the intersection of any two edges of different hypergraphs cannot consist of a prescribed number of vertices. We also present corollaries to these results in the problem of finding the chromatic number of a space with a forbidden equilateral triangle with monochromatic vertices.