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

Let µ (G) be the largest eigenvalue of a graph G and Tr (n) be the r-partite Turán graph of order n. We prove that if G is a graph of order n with µ (G)> µ (Tr (n)) , then G contains various large supergraphs of the complete graph of order r + 1, e.g., the complete r-partite graph with all parts of size log n with an edge added to the first part. We also give corresponding stability results.

Let µ (G) be the largest eigenvalue of a graph G, let Kr (s1,...,sr) be the complete rpartite graph with parts of size s1,...,sr, and let Tr (n) be the r-partite Turán graph of order n. Our main result is: For all r ≥ 2 and all sufficiently small c> 0, ε> 0, every graph G of sufficiently large order n with µ (G) ≥ (1 − 1/r −�ε)n satisfies one of the � conditions: (a) G contains a Kr+1 ⌊cln n⌋,..., ⌊cln n⌋, n1− √ �� c; (b) G differs from Tr (n) in fewer than � ε 1/4 + c 1/(8r+8) � n 2 edges. In particular, this result strengthens the stability theorem of Erdős and Simonovits.

Let K + r (s1,...,sr) be the complete r-partite graph with parts of size s1 ≥ 2,s2,...,sr with an edge added to the first part. Letting tr (n) be the number of edges of the r-partite Turán graph of order n, we prove that: (A) For all r ≥ 2 and all sufficiently small ε> 0, every graph of sufficiently large order n with tr (n) + 1 edges contains a K + r ⌊cln n⌋,..., ⌊cln n⌋, n 1− √ c (B) For all r ≥ 2, there exists c> 0 such that every graph of sufficiently large order n with tr (n) + 1 edges contains a K + r (⌊cln n⌋,..., ⌊cln n⌋). These assertions extend results of Erdős from 1963. We also give corresponding stability results Keywords: clique; r-partite graph; stability, Turán’s theorem