A set of q triangles sharing a common edge is called a book of size q. We write β (n, m) for the the maximal q such that every graph G (n, m) contains a book of size q. In this note 1) we compute β ( n, cn 2) for infinitely many values of c with 1/4 < c < 1/3, 2) we show that if m ≥ (1/4 − α) n 2 with 0 < α < 17 −3 (), and G has no book of size at least graph G1 of order at least

Given a graph G with II vertices and m edges, how many edges must be in thelargest chordal subgraph of G? For m=na/4+ 1, the answer is 3n/2- 1. For m=na/3, it is 2n-3. For m = n2/3 + 1, it is at least 7n/3- 6 and at most 8n/3- 4. Similar questions are studied, with less complete resuIts, for threshold graphs, interval graphs, and the stars on edges, triangles, and &‘s.