Books in graphs

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 ) for infinitely many values of c with 1/4 < c < 1/3, 2) we show that if m ≥ (1/4− α)n with 0 < α < 17, and G has no book of size at least ( 1/6 − 2α ) n then G contains an induced bipartite graph G1 of order at least ( 1− α ) n and minimal degree δ (G1) ≥ ( 1 2 − 4α ) n, 3) we apply the latter result to answer two questions of Erdős concerning the booksize of graphs G ( n, n/4− f (n)n ) every edge of which is contained in a triangle, and 0 < f (n) < n.