For a random permutation of $n$ objects, as $n \to \infty$, the process giving the proportion of elements in the longest cycle, the second-longest cycle, and so on, converges in distribution to the Poisson–Dirichlet process with parameter 1. This was proved in 1977 by Kingman and by Vershik and Schmidt. For soft reasons, this is equivalent to the statement that the random permutations and the Poisson–Dirichlet process can be coupled so that zero is the limit of the expected $\ell_1$ distance between the process of cycle length proportions and the Poisson–Dirichlet process. We investigate how rapid this metric convergence can be, and in doing so, give two new proofs of the distributional convergence.

One of the couplings we consider has an analogue for the prime factorizations of a uniformly distributed random integer, and these couplings rely on the ‘scale-invariant spacing lemma’ for the scale-invariant Poisson processes, proved in this paper.

We put the final piece into a puzzle first introduced by Bollobás, Erdõs and Szemerédi in 1975. For arbitrary positive integers $n$ and $r$ we determine the largest integer $\Delta=\Delta (r,n)$, for which any $r$-partite graph with partite sets of size $n$ and of maximum degree less than $\Delta$ has an independent transversal. This value was known for all even $r$. Here we determine the value for odd $r$ and find that $\Delta(r,n)=\Delta(r-1,n)$. Informally this means that the addition of an oddth partite set does not make it any harder to guarantee an independent transversal.

In the proof we establish structural theorems which could be of independent interest. They work for all$r\geq 7$, and specify the structure of slightly sub-optimal graphs for even$r\geq 8$.

We show that a maximum cut of a random graph below the giant-component threshold can be found in linear space and linear expected time by a simple algorithm. In fact, the algorithm solves a more general class of problems, namely binary 2-variable constraint satisfaction problems. In addition to Max Cut, such Max 2-CSPs encompass Max Dicut, Max 2-Lin, Max 2-Sat, Max-Ones-2-Sat, maximum independent set, and minimum vertex cover. We show that if a Max 2-CSP instance has an ‘underlying’ graph which is a random graph $G \in \mathcal{G}(n,c/n)$, then the instance is solved in linear expected time if $c \leq 1$. Moreover, for arbitrary values (or functions) $c>1$ an instance is solved in expected time $n \exp(O(1+(c-1)^3 n))$; in the ‘scaling window’ $c=1+\lambda n^{-1/3}$ with $\lambda$ fixed, this expected time remains linear.

Our method is to show, first, that if a Max 2-CSP has a connected underlying graph with $n$ vertices and $m$ edges, then $O(n 2^{(m-n)/2})$ is a deterministic upper bound on the solution time. Then, analysing the tails of the distribution of this quantity for a component of a random graph yields our result. Towards this end we derive some useful properties of binomial distributions and simple random walks.

We study relational structures (especially graphs and posets) which satisfy the analogue of homogeneity but for homomorphisms rather than isomorphisms. The picture is rather different. Our main results are partial characterizations of countable graphs and posets with this property; an analogue of Fraïssé's theorem; and representations of monoids as endomorphism monoids of such structures.

In this paper, we give sharp upper bounds on the maximum number of edges in very unbalanced bipartite graphs not containing any cycle of length 6. To prove this, we estimate roughly the sum of the sizes of the hyperedges in triangle-free multi-hypergraphs.

The adaption of combinatorial duality to infinite graphs has been hampered by the fact that while cuts (or cocycles) can be infinite, cycles are finite. We show that these obstructions fall away when duality is reinterpreted on the basis of a ‘singular’ approach to graph homology, whose cycles are defined topologically in a space formed by the graph together with its ends and can be infinite. Our approach enables us to complete Thomassen's results about ‘finitary’ duality for infinite graphs to full duality, including his extensions of Whitney's theorem.

In this paper, I give a short proof of a recent result by Sokal, showing that all zeros of the chromatic polynomial $P_G(q)$ of a finite graph $G$ of maximal degree $D$ lie in the disk $|q|< K D$, where $K$ is a constant that is strictly smaller than 8.

In the present work we prove the following conjecture of Erdős, Roth, Sárközy and T. Sós: Let $f$ be a polynomial of integer coefficients such that $2|f(z)$ for some integer $z$. Then, for any $k$-colouring of the integers, the equation $x+y=f(z)$ has a solution in which $x$ and $y$ have the same colour. A well-known special case of this conjecture referred to the case $f(z)=z^2$.

Mauduit and Sárközy introduced and studied certain numerical parameters associated to finite binary sequences $E_N\in\{-1,1\}^N$ in order to measure their ‘level of randomness’. Two of these parameters are the normality measure$\cal{N}(E_N)$ and the correlation measure$C_k(E_N)$of order k, which focus on different combinatorial aspects of $E_N$. In their work, amongst others, Mauduit and Sárközy investigated the minimal possible value of these parameters.

In this paper, we continue the work in this direction and prove a lower bound for the correlation measure $C_k(E_N)$ (k even) for arbitrary sequences $E_N$, establishing one of their conjectures. We also give an algebraic construction for a sequence $E_N$ with small normality measure $\cal{N}(E_N)$.