Suppose that n > (log k)ck, where c is a fixed positive constant. We prove that, no matter how the edges of Kn are coloured with k colours, there is a copy of K4 whose edges receive at most two colours. This improves the previous best bound of kc′k, where c′ is a fixed positive constant, which follows from results on classical Ramsey numbers.

Starting from a sequence regarded as a walk through some set of values, we consider the associated loop-erased walk as a sequence of directed edges, with an edge from i to j if the loop-erased walk makes a step from i to j. We introduce a colouring of these edges by painting edges with a fixed colour as long as the walk does not loop back on itself, then switching to a new colour whenever a loop is erased, with each new colour distinct from all previous colours. The pattern of colours along the edges of the loop-erased walk then displays stretches of consecutive steps of the walk left untouched by the loop-erasure process. Assuming that the underlying sequence generating the loop-erased walk is a sequence of independent random variables, each uniform on [N] := {1, 2, . . ., N}, we condition the walk to start at N and stop the walk when it first reaches the subset [k], for some 1 ≤ k ≤ N − 1. We relate the distribution of the random length of this loop-erased walk to the distribution of the length of the first loop of the walk, via Cayley's enumerations of trees, and via Wilson's algorithm. For fixed N and k, and i = 1, 2, . . ., let Bi denote the events that the loop-erased walk from N to [k] has i + 1 or more edges, and the ith and (i + 1)th of these edges are coloured differently. We show that, given that the loop-erased random walk has j edges for some 1 ≤ j ≤ N − k, the events Bi for 1 ≤ i ≤ j − 1 are independent, with the probability of Bi equal to 1/(k + i + 1). This determines the distribution of the sequence of random lengths of differently coloured segments of the loop-erased walk, and yields asymptotic descriptions of these random lengths as N → ∞.

In this paper we provide a simple formula for the expected time for a random recursive tree to grow to a given height.

A set of integers is called a B2[g] set if every integer m has at most g representations of the form m = a + a′, with a ≤ a′ and a, a′ ∈ . We obtain a new lower bound for F(g, n), the largest cardinality of a B2[g] set in {1,. . .,n}. More precisely, we prove that infn→∞ where ϵg → 0 when g → ∞. We show a connection between this problem and another one discussed by Schinzel and Schmidt, which can be considered its continuous version.

We study the asymptotic distribution of the displacements in hashing with coalesced chains, for both late-insertion and early-insertion. Asymptotic formulas for means and variances follow. The method uses Poissonization and some stochastic calculus.

Gerards and Seymour (see [10], p. 115) conjectured that if a graph has no odd complete minor of order l, then it is (l − 1)-colourable. This is an analogue of the well-known conjecture of Hadwiger, and in fact, this would immediately imply Hadwiger's conjecture. The current best-known bound for the chromatic number of graphs with no odd complete minor of order l is by the recent result by Geelen, Gerards, Reed, Seymour and Vetta [8], and by Kawarabayashi [12] later, independently. But it seems very hard to improve this bound since this would also improve the current best-known bound for the chromatic number of graphs with no complete minor of order l.

Motivated by this problem, in this note we show that there exists an absolute constant f(k) such that any graph G with no odd complete minor of order k admits a vertex partition V1, . . ., V496k such that each component in the subgraph induced on Vi (i ≥ 1) has at most f(k) vertices. When f(k) = 1, this is a colouring of G. Hence this is a relaxation of colouring in a sense, and this is the first result in this direction for the odd Hadwiger's conjecture.

Our proof is based on a recent decomposition theorem due to Geelen, Gerards, Reed, Seymour and Vetta [8], together with a connectivity result that forces a huge complete bipartite minor in large graphs by Böhme, Kawarabayashi, Maharry and Mohar [3].

Sokal in 2001 proved that the complex zeros of the chromatic polynomial PG(q) of any graph G lie in the disc |q| < 7.963907Δ, where Δ is the maximum degree of G. This result answered a question posed by Brenti, Royle and Wagner in 1994 and hence proved a conjecture proposed by Biggs, Damerell and Sands in 1972. Borgs gave a short proof of Sokal's result. Fernández and Procacci recently improved Sokal's result to |q| < 6.91Δ. In this paper, we shall show that all real zeros of PG(q) are in the interval [0,5.664Δ). For the special case that Δ = 3, all real zeros of PG(q) are in the interval [0,4.765Δ).

A graph construction game is a Maker–Breaker game. Maker and Breaker take turns in choosing previously unoccupied edges of the complete graph KN. Maker's aim is to claim a copy of a given target graph G while Breaker's aim is to prevent Maker from doing so. In this paper we show that if G is a d-degenerate graph on n vertices and N > d1122d+9n, then Maker can claim a copy of G in at most d1122d+7n rounds. We also discuss a lower bound on the number of rounds Maker needs to win, and the gap between these bounds.