On graphs which contain all small trees

Communicated by the Managing Editors We investigate those graphs G, with the property that any tree on N vertices occurs as subgraph of G,. In particular, we consider the problem of estimating the minimum number of edges such a graph can have. We show that this number is bounded below and above by $z log II and nl+l/log log %, respectively. A typical question in extremal graph theory1 is one which asks for the maximum number of edges a graph G on n vertices can have so that G does not contain some given graph H (or class of graphs Z) as a subgraph. Perhaps the most well-known result of this type is the theorem of Turdn [9, lo] which asserts that if H = K,,, , the complete graph on m vertices, then this maximum number is just where r is the unique integer satisfying r = Iz(mod n?-1) and l,<I.<:M?-1.