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

A subgraph of a graph G is called trivial if it is either a clique or an independent set. Let qG denote the maximum number of vertices in a trivial subgraph of G. Motivated by an open problem of Erdo s and McKay we show that every graph G on n vertices for which qG#C log n contains an induced subgraph with exactly y edges, for every y between 0 and n #C . Our methods enable us also to show that under much weaker assumption, i.e., qG#n=14, G still must contain an induced subgraph with exactly y edges, for every y between 0 and e .

Abstract not found

For a real c ≥ 1 and an integer n, let f(n, c) denote the maximum integer f such that every graph on n vertices contains an induced subgraph on at least f vertices in which the maximum degree is at most c times the minimum degree. Thus, in particular, every graph on n vertices contains a regular induced subgraph on at least f(n,1) vertices. The problem of estimating f(n,1) was posed long ago by Erdős, Fajtlowicz, and Staton. In this paper we obtain the following upper and lower bounds for the asymptotic behavior of f(n, c): (i) For fixed c>2.1, n 1−O(1/c) ≤ f(n, c) ≤ O(cn / log n). (ii) For fixed c =1+ε with ε>0 sufficiently small, f(n, c) ≥ n Ω(ε2 / ln(1/ε)). (iii) Ω(ln n) ≤ f(n, 1) ≤ O(n 1/2 ln 3/4 n). An analogous problem for not necessarily induced subgraphs is briefly considered as well.

Abstract not found

We prove that every Hamiltonian graph with n vertices and m edges has cycles of 4 at least

induced subgraphs of a random graph

We describe a simple and yet surprisingly powerful probabilistic technique which shows how to find in a dense graph a large subset of vertices in which all (or almost all) small subsets have many common neighbors. Recently this technique has had several striking applications to Extremal Graph Theory, Ramsey Theory, Additive Combinatorics, and Combinatorial Geometry. In this survey we discuss some of them. 1