Every matroid is a submatroid of a uniformly dense matroid (Q1902898)

For a graph \(G\) with at least one edge, define \(d(G)= {|E(G)|\over |V(G)|}\) and \(m(G)= \max\{d(H)\mid H\subseteq G\}\). Györi proved that every connected graph is a subgraph of a graph \(G'\) with \(m(G')= d(G)= m(G)\). A similar extremal problem was investigated by Payan. The authors in this paper show that both problems (and theorems) above are related by matroid elongations. They extend these results to their versions of binary and regular matroids.