An instance of the square packing problem consists of n squares with independently, uniformly distributed side-lengths and independently, uniformly distributed locations on the unit d-dimensional torus. A packing is a maximum family of pairwise disjoint squares. The one-dimensional version of the problem is the classical random interval packing problem. This paper deals with the asymptotic behaviour of packings as n tends to infinity while d = 2. Coffman, Lueker, Spencer and Winkler recently proved that the average size of packing is Θ(nd/(d+1)). Using partitioning techniques, sub-additivity and concentration of measure arguments, we show first that, after normalization by n2/3, the size of two-dimensional square packings tends in probability toward a genuine limit γ. Straightforward concentration arguments show that large fluctuations of order n2/3 should have probability vanishing exponentially fast with n2/3. Even though γ remains unknown, using a change of measure argument we show that this upper bound on tail probabilities is qualitatively correct.

We define a space of random edge-coloured graphs [Gscr ]n,m,κ which correspond naturally to edge κ-colourings of Gn,m. We show that there exist constants K0, K1 [les ] 21 such that, provided m [ges ] K0n log n and κ [ges ] K1n, then a random edge-coloured graph contains a multi-coloured Hamilton cycle with probability tending to 1 as the number of vertices n tends to infinity.

We consider a stochastic process based on the iterated prisoner's dilemma game. During the game, each of n players has a state, either cooperate or defect. The players are connected by an ‘interaction graph’. During each step of the process, an edge of the graph is chosen uniformly at random and the states of the players connected by the edge are modified according to the Pavlov strategy. The process converges to a unique absorbing state in which all players cooperate. We prove two conjectures of Kittock: the convergence rate is exponential in n when the interaction graph is a complete graph, and it is polynomial in n when the interaction graph is a cycle. In fact, we show that the rate is O(n log n) in the latter case.

Let Tn be the complete binary tree of height n considered as the Hasse diagram of a poset with its root 1n as the maximum element. Define A(n; T) = [mid ]{S ⊆ Tn : 1n ∈ S, S ≅ T}[mid ], and B(n; T) = [mid ]{S ⊆ Tn : 1n ∉ S, S ≅ T}[mid ]. In this note we prove that for any fixed n and rooted binary trees T1, T2 such that T2 contains a subposet isomorphic to T1. We conjecture that the ratio A/B also increases with T for arbitrary trees. These inequalities imply natural behaviour of the optimal stopping time in a poset extension of the secretary problem.

We give a concentration inequality involving a family of independent random permutations, which is useful for analysing certain randomized methods for graph colouring.

Let λ(G) be the largest eigenvalue of the adjacency matrix of a graph G: We show that if G is Kp+1-free then

This inequality was first conjectured by Edwards and Elphick in 1983 and supersedes a series of previous results on upper bounds of λ(G).

Let Ti denote the number of all i-cliques of G, λ = λ(G) and p = cl(G): We show

Let δ be the minimal degree of G. We show

This inequality supersedes inequalities of Stanley and Hong. It is sharp for regular graphs and for a class of graphs which are in some sense maximally irregular.

We show that, for every positive integer c*, there is an integer n such that, if M is a matroid whose largest cocircuit has size c*, then E(M) can be partitioned into two sets E1 and E2 such that every connected component of each of M[mid ]E1 and M[mid ]E2 has at most n elements.

How large can the Lagrangian of an r-graph with m edges be? Frankl and Füredi [1] conjectured that the r-graph of size m formed by taking the first m sets in the colex ordering of N(r) has the largest Lagrangian of all r-graphs of size m. We prove the first ‘interesting’ case of this conjecture, namely that the 3-graph with (t3) edges and largest Lagrangian is [t](3). We also prove that this conjecture is true for 3-graphs of several other sizes.

For general r-graphs we prove a weaker result: for t sufficiently large, the r-graph of size (tr) supported on t + 1 vertices and with largest Lagrangian, is [t](r).