Denote by fn(H) the number of (labeled) H-free graphs on a fixed vertex set of size n. Erdős conjectured that whenever H contains a cycle, fn(H) = 2 (1+o(1)) ex(n,H) , yet it still open for every bipartite graph, and even the order of magnitude of log 2 fn(H) was known only for C4, C6 and K3,3. We show that for all 2 ≤ s ≤ t, fn(Ks,t) = 2 O(n2−1/s) which is asymptotically sharp for all those values of s and t for which the order of magnitude of the Turán number ex(n, Ks,t) is known. Our methods allow us to prove, among other things, that there is a positive constant c, such that almost all K2,t-free graphs of order n have at least 1/12 · ex(n, K2,t) and at most (1 − c) · ex(n, K2,t) edges. Moreover, our results have some interesting applications to the study of some Ramseyand Turán-type problems. 1

Following the decontamination metaphor for searching a graph, we introduce a cleaning process, which is related to both the chip-firing game and edge searching. Brushes (instead of chips) are placed on some vertices and, initially, all the edges are dirty. When a vertex is ‘fired’, each dirty incident edge is traversed by only one brush, cleaning it, but a brush is not allowed to traverse an already cleaned edge; consequently, a vertex may not need degree-many brushes to fire. The model presented is one where the edges are continually recontaminated, say by algae, so that cleaning is regarded as an on-going process. Ideally, the final configuration of the brushes, after all the edges have been cleaned, should be a viable starting configuration to clean the graph again. We show that this is possible with the least number of brushes if the vertices are fired sequentially but not if fired in parallel. We also present bounds for the least number of brushes required to clean graphs in general and some specific families of graphs.

The investigation of the asymptotic behaviour of various parameters of powers of a fixed graph leads to many fascinating problems, some of which are motivated by questions in information theory, communication complexity, geometry and Ramsey theory. In this survey we discuss these problems and describe the techniques used in their study which combine combinatorial, geometric, probabilistic and linear-algebra tools. 1 Graph Powers There are several known distinct ways to define the powers of a fixed graph. The n-th AND power of an undirected graph G = (V, E) is the graph denoted by G ∧n whose vertex set is V n in which distinct vertices (x1... xn) and (x ′ 1... x ′ n) are connected if {xi, x ′ i} ∈ E for all i ∈ {1, 2,..., n} such that xi � = x ′ i. The n-th OR power of G is the graph denoted G∨n whose vertex set is V n in which distinct vertices (x1... xn) and (x ′ 1... x ′ n) are connected if distinct xi and x ′ i for some i ∈ {1, 2,..., n}. are connected in G The study of the asymptotic behaviour of various parameters of these powers of a fixed graph G, as well as their behaviour for similarly defined powers of directed and undirected graphs, is motivated by questions in various areas and leads to many intriguing problems. These are discussed in the following sections, in which we focus our attention mainly to the open problems in the area, and only briefly describe the known results and proof techniques. Proven and disproven conjectures are intermingled throughout the paper with open problems. More detailed proofs can be found in the papers listed in the bibliography. 2 Shannon Capacity The independence number α(G) of a graph G is the maximum cardinality of a set of vertices of G no two of which are adjacent.

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A graph is called H-free if it contains no copy of H. An old result of Kleitman and Winston [12] states that there are 2 Θ(n3/2) C4-free graphs on n vertices. Füredi [8] showed that almost all C4-free graphs of order n have at least c · ex(n, C4) edges for some positive constant c> 0. We prove that there is an ε> 0 such that almost all C4-free graphs have at most (1 − ε) · ex(n, C4) edges. This resolves a conjecture of Balogh, Bollobás and Simonovits [4] for the 4-cycle. Mathematics subject classification: 05C35, 05C30, 05D40, 05A16 Keywords: asymptotic graph structure; asymptotic graph enumeration; C4-free; Turán’s problem; extremal graphs 1

We establish the following bounds on the star-vs-triangle size ramsey number. n 2 + (0:577 + o(1)) n 3=2 ! r(K 1;n ; K 3 ) ! n 2 + p 2 n 3=2 + n: The upper (constructive) bound disproves a conjecture of Erdos. Introduction A conjecture of Erdos [3], see also e.g. [2, 4], states that for n 3 any graph with \Gamma 2n+1 2 \Delta \Gamma \Gamma n 2 \Delta \Gamma 1 edges is a union of a bipartite graph and a graph with maximum degree less than n. This value arises from the consideration of P n+1;n which doesn't admit the above representation. (P m;n = Km+E n has m+ n vertices of which m vertices are connected to every other vertex.) In the arrowing notation the latter statement reads "P n+1;n ! (K 1;n ; C odd )": for any blue-red colouring of the edge-set of P n+1;n we necessarily have either a blue star K 1;n or a red cycle of odd length. (By C odd we denote the family of odd cycles.) Thus the conjecture states that r(K 1;n ; C odd ) = e(P n+1;n ) and, if true, g...

. Let m(n; n + k) be the number of connected labelled multigraphs, which are graphs with n vertices, n + k edges and possible selfloops and/or multiple edges. Denote by c(n; n + k) the number of connected labelled simple graphs with the same parameters. First, under the condition that k = o(n 2 ), by making use of the methods developped by Bender et al. in [3], we show that m(n;n + k) c(n; n + k) as n ! 1. Under the same condition on the number of exceeding edges, k = o(n 2 ), these results are extended to show that connected labelled graphs, multigraphs and graphs without a nite set of forbidden subgraphs have the same asymptotic behavior. Finally, we give sucient condition, in terms of the total number of graphs, for the probability of connectedness to have a limit equal to 1 as the number of vertices tends to 1. 1 Introduction In this paper, all graphs are labelled and simple, i.e., without self-loops or multiple edges. There is a large body of research devoted to the enum...

A graph is called H-free if it contains no copy of H. Denote by fn(H) the number of (labeled) H-free graphs on n vertices. Erdős conjectured (see [7]) that fn(H) ≤ 2 (1+o(1)) ex(n,H). This was first shown to be true for cliques [9] and then Erdős, Frankl and Rödl [8] proved it for all graphs H with χ(H) ≥ 3. For most bipartite H, the question is still wide open, and even the correct order of magnitude of log 2 fn(H) is not known. We prove that for every m ≥ 2, fn(Km,m) ≤ 2 O(n2−1/m), extending the result of Kleitman and Winston [15] and answering a question of Erdős. This bound is asymptotically sharp for m ∈ {2, 3}, and possibly for other values of m, for which the order of ex(n, Km,m) is not known, but it is conjectured to be Θ(n 2−1/m). Our method also yields a bound on the number of Km,m-free graphs with fixed order and size, extending the result of Füredi [11]. Using this bound, we prove a relaxed version of a conjecture due to Haxell, Kohayakawa and ̷Luczak [13] and show that almost all K3,3-free graphs of order n have more than 1/20 · ex(n, K3,3) edges. 1

“Aliens invade the earth and threaten to obliterate it in a year’s time unless human beings can find the Ramsey number for red five and blue five. We could marshal the world’s best minds and fastest computers, and within a year we could probably calculate the value. If the aliens demanded the Ramsey number for red six and blue six, however, we would have no choice but to launch a preemptive attack.” Paul Erdős [5] In this paper, we consider the on-line Ramsey numbers R(k, l) for cliques. Using a high performance computing networks, we ‘calculated ’ that R(3, 4) = 17. We also present an upper bound of R(k, l), study its asymptotic behaviour, and state some open problems. 1 Introduction and