Frankl, Füredi, Győri and Pach obtained fundamental results in extremal combinatorics, graph theory and discrete geometry. In this paper we describe solutions or partial solutions of several problems in these areas. The problems considered deal with bipartite coverings of graphs, triangle-free graphs of diameter 2, and geometric and combinatorial designs and partial designs. This is a (short) sequel of several previous papers of the same flavour. Each section here is essentially self contained, and can be read separately.