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

We prove a number of relations between the number of cliques of a graph G and the largest eigenvalue µ (G) of its adjacency matrix. In particular, writing ks (G) for the number of s-cliques of G, we show that, for all r ≥ 2, µ r+1 (G) ≤ (r + 1) kr+1 (G) + and, if G is of order n, then µ (G) kr+1 (G) ≥ n r� (s − 1) ks (G) µ r+1−s (G), s=2 1 r (r − 1)

For a fixed ρ ∈ [0, 1], what is (asymptotically) the minimal possible density g3(ρ) of triangles in a graph with edge density ρ? We completely solve this problem by proving that (t − 1) t − 2 g3(ρ) = t(t − ρ(t + 1)) t + � �2 t(t − ρ(t + 1)) t2 (t + 1) 2 where t def

In 1969 Erdős proved that if r ≥ 2 and n> n0 (r) , every graph G of order n and e (G)> tr (n) has an edge that is contained in at least n r−1 /(10r) 6r (r + 1)-cliques. In this note we improve this bound to n r−1 /r r+5. We also prove a corresponding stability result.

Graphs with many r-cliques have large complete

Graphs with many r-cliques have large complete r-partite subgraphs