Let C be a convex cone in ℛd with non-empty interior and a compact basis K. If H1 and H2 are any two parallel hyperplanes tangent to K, whose slices with C are two other compact basis K1 and K2, let D, D1 and D2 be the truncated subcones of C generated by K, K1 and K2. We prove that K is an ellipsoid if, and only if, vol (D)2 = vol (D1) vol (D2) for every such pair of hyperplanes H1, and H2.

We prove that in every finite dimensional normed space, for “most” pairs (x, y) of points in the unit ball, ║x − y║ is more than √2(1 − ε). As a consequence, we obtain a result proved by Bourgain, using QS-decomposition, that guarantees an exponentially large number of points in the unit ball any two of which are separated by more than √2(1 − ε).

We consider continuous time random walks on a product graph G×H, where G is arbitrary and H consists of two vertices x and y linked by an edge. For any t>0 and any a, b∈V(G), we show that the random walk starting at (a, x) is more likely to have hit (b, x) than (b, y) by time t. This contrasts with the discrete time case and proves a conjecture of Bollobás and Brightwell. We also generalize the result to cases where H is either a complete graph on n vertices or a cycle on n vertices.

For any integer k, we prove the existence of a uniquely k-colourable graph of girth at least g on at most k12(g+1) vertices whose maximal degree is at most 5k13. From this we deduce that, unless NP=RP, no polynomial time algorithm for k-Colourability on graphs G of girth g(G)[ges ]log[mid ]G[mid ]/13logk and maximum degree Δ(G)[les ]6k13 can exist. We also study several related problems.

We consider the function χ(Gk), defined to be the smallest number of colours that can colour a graph G in such a way that no vertices of distance at most k receive the same colour. In particular we shall look at how small a value this function can take in terms of the order and diameter of G. We get general bounds for this and tight bounds for the cases k=2 and k=3.

For any Boolean function f, let L(f) be its formula size complexity in the basis {∧, [oplus ] 1}. For every n and every k[les ]n/2, we describe a probabilistic distribution on formulas in the basis {∧, [oplus ] 1} in some given set of n variables and of size at most [lscr](k)=4k. Let pn,k(f) be the probability that the formula chosen from the distribution computes the function f. For every function f with L(f)[les ][lscr](k)α, where α=log4(3/2), we have pn,k(f)>0. Moreover, for every function f, if pn,k(f)>0, then

formula here

where c>1 is an absolute constant. Although the upper and lower bounds are exponentially small in [lscr](k), they are quasi-polynomially related whenever [lscr](k)[ges ]lnΩ(1)n. The construction is a step towards developing a model appropriate for investigation of the properties of a typical (random) Boolean function of some given complexity.

We prove that there exists n0, such that, for every n[ges ]n0 and every 2-colouring of the edges of the complete graph Kn, one can find two vertex-disjoint monochromatic cycles of different colours which cover all vertices of Kn.

To bound the probability of a union of n events from a single set of events, Bonferroni inequalities are sometimes used. There are sharper bounds which are called Sobel–Uppuluri–Galambos inequalities. When two (or more) sets of events are involved, bounds are considered on the probability of intersection of several such unions, one union from each set. We present a method for unified treatment of bivariate lower and upper bounds in this note. The lower bounds obtained are new and at least as good as lower bounds appearing in the literature so far. The upper bounds coincide with existing bivariate Sobel–Uppuluri–Galambos type upper bounds derived by the method of indicator functions. A numerical example is given to illustrate that the new lower bounds can be strictly better than existing ones.

Given a sequence of nonnegative real numbers λ0, λ1, … that sum to 1, we consider a random graph having approximately λin vertices of degree i. In [12] the authors essentially show that if [sum ]i(i−2)λi>0 then the graph a.s. has a giant component, while if [sum ]i(i−2)λi<0 then a.s. all components in the graph are small. In this paper we analyse the size of the giant component in the former case, and the structure of the graph formed by deleting that component. We determine ε, λ′0, λ′1 … such that a.s. the giant component, C, has εn+o(n) vertices, and the structure of the graph remaining after deleting C is basically that of a random graph with n′=n−[mid ]C[mid ] vertices, and with λ′in′ of them of degree i.