The inducibility of a graph H measures the maximum number of induced copies of H a large graph G can have. Generalizing this notion, we study how many induced subgraphs of fixed order k and size ℓ a large graph G on n vertices can have. Clearly, this number is $\left( {\matrix{n \cr k}}\right)$ for every n, k and $\ell \in \left\{ {0,\left( {\matrix{k \cr 2}} \right)}\right\}$. We conjecture that for every n, k and $0 \lt \ell \lt \left( {\matrix{k \cr 2}}\right)$ this number is at most $ (1/e + {o_k}(1)) {\left( {\matrix{n \cr k}} \right)}$. If true, this would be tight for ℓ ∈ {1, k − 1}.

In support of our ‘Edge-statistics Conjecture’, we prove that the corresponding density is bounded away from 1 by an absolute constant. Furthermore, for various ranges of the values of ℓ we establish stronger bounds. In particular, we prove that for ‘almost all’ pairs (k, ℓ) only a polynomially small fraction of the k-subsets of V(G) have exactly ℓ edges, and prove an upper bound of $ (1/2 + {o_k}(1)){\left( {\matrix{n \cr k}}\right)}$ for ℓ = 1.

Our proof methods involve probabilistic tools, such as anti-concentration results relying on fourth moment estimates and Brun’s sieve, as well as graph-theoretic and combinatorial arguments such as Zykov’s symmetrization, Sperner’s theorem and various counting techniques.

For positive integers n and q and a monotone graph property $\mathcal{A}$, we consider the two-player, perfect information game WC(n, q, $\mathcal{A}$), which is defined as follows. The game proceeds in rounds. In each round, the first player, called Waiter, offers the second player, called Client, q + 1 edges of the complete graph Kn which have not been offered previously. Client then chooses one of these edges which he keeps and the remaining q edges go back to Waiter. If, at the end of the game, the graph which consists of the edges chosen by Client satisfies the property $\mathcal{A}$, then Waiter is declared the winner; otherwise Client wins the game. In this paper we study such games (also known as Picker–Chooser games) for a variety of natural graph-theoretic parameters, such as the size of a largest component or the length of a longest cycle. In particular, we describe a phase transition type phenomenon which occurs when the parameter q is close to n and is reminiscent of phase transition phenomena in random graphs. Namely, we prove that if q ⩾ (1 + ϵ)n, then Client can avoid components of order cϵ−2 ln n for some absolute constant c > 0, whereas for q ⩽ (1 − ϵ)n, Waiter can force a giant, linearly sized component in Client's graph. In the second part of the paper, we prove that Waiter can force Client's graph to be pancyclic for every q ⩽ cn, where c > 0 is an appropriate constant. Note that this behaviour is in stark contrast to the threshold for pancyclicity and Hamiltonicity of random graphs.

We study 3-random-like graphs, that is, sequences of graphs in which the densities of triangles and anti-triangles converge to 1/8. Since the random graph $\mathcal{G}$n,1/2 is, in particular, 3-random-like, this can be viewed as a weak version of quasi-randomness. We first show that 3-random-like graphs are 4-universal, that is, they contain induced copies of all 4-vertex graphs. This settles a question of Linial and Morgenstern [10]. We then show that for larger subgraphs, 3-random-like sequences demonstrate completely different behaviour. We prove that for every graph H on n ⩾ 13 vertices there exist 3-random-like graphs without an induced copy of H. Moreover, we prove that for every ℓ there are 3-random-like graphs which are ℓ-universal but not m-universal when m is sufficiently large compared to ℓ.

Let P be a monotone increasing graph property, let G = (V, E) be a graph, and let q be a positive integer. In this paper, we study the (1: q) Maker–Breaker game, played on the edges of G, in which Maker's goal is to build a graph that satisfies the property P. It is clear that in order for Maker to have a chance of winning, G itself must satisfy P. We prove that if G satisfies P in some strong sense, that is, if one has to delete sufficiently many edges from G in order to obtain a graph that does not satisfy P, then Maker has a winning strategy for this game. We also consider a different notion of satisfying some property in a strong sense, which is motivated by a problem of Duffus, Łuczak and Rödl [6].