Uniform Families and Count Matroids

Abstract.

Given an r-graph G on [n], we are allowed to add consecutively new edges to it provided that every time a new r-graph with at least l edges and at most m vertices appears. Suppose we have been able to add all edges. What is the minimal number of edges in the original graph? For all values of parameters, we present an example of G which we conjecture to be extremal and establish the validity of our conjecture for a range of parameters. Our proof utilises count matroids which is a new family of matroids naturally extending that of White and Whiteley. We characterise, in certain cases, the extremal graphs. In particular, we answer a question by Erdős, Füredi and Tuza.