Zeilberger's enumeration schemes can be used to completely automate the enumeration of many permutation classes. We extend his enumeration schemes so that they apply to many more permutation classes and describe the Maple package WilfPlus, which implements this process. We also compare enumeration schemes to three other systematic enumeration techniques: generating trees, substitution decompositions, and the insertion encoding.

If G is a graph with vertex set [n] then is G-intersecting if, for all , either A ∩ B ≠ ∅ or there exist a ∈ A and b ∈ B such that a ~Gb.

The question of how large a k-uniform G-intersecting family can be was first considered by Bohman, Frieze, Ruszinkó and Thoma [2], who identified two natural candidates for the extrema depending on the relative sizes of k and n and asked whether there is a sharp phase transition between the two. Our first result shows that there is a sharp transition and characterizes the extremal families when G is a matching. We also give an example demonstrating that other extremal families can occur.

Our second result gives a sufficient condition for the largest G-intersecting family to contain almost all k-sets. In particular we show that if Cn is the n-cycle and k > αn + o(n), where α = 0.266. . . is the smallest positive root of (1 − x)3(1 + x) = 1/2, then the largest Cn-intersecting family has size .

Finally we consider the non-uniform problem, and show that in this case the size of the largest G-intersecting family depends on the matching number of G.

We analyse classes of planar graphs with respect to various properties such as polynomial-time solvability of the dominating set problem or boundedness of the tree-width. A helpful tool to address this question is the notion of boundary classes. The main result of the paper is that for many important properties there are exactly two boundary classes of planar graphs.

The following two conjectures arose in the work of Grimmett and Winkler, and Pemantle: the uniformly random forest F and the uniformly random connected subgraph C of a finite graph G have the edge-negative association property. In other words, for all distinct edges e and f of G, the probability that F (respectively, C) contains e conditioned on containing f is less than or equal to the probability that F (respectively, C) contains e. Grimmett and Winkler showed that the first conjecture is true for all simple graphs on 8 vertices and all graphs on 9 vertices with at most 18 edges. In this paper, we describe an infinite, nontrivial class of graphs and matroids for which a generalized version of both conjectures holds.

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].

For a graph G and an integer t we let mcct(G) be the smallest m such that there exists a colouring of the vertices of G by t colours with no monochromatic connected subgraph having more than m vertices. Let be any non-trivial minor-closed family of graphs. We show that mcc2(G) = O(n2/3) for any n-vertex graph G ∈ . This bound is asymptotically optimal and it is attained for planar graphs. More generally, for every such , and every fixed t we show that mcct(G)=O(n2/(t+1)). On the other hand, we have examples of graphs G with no Kt+3 minor and with mcct(G)=Ω(n2/(2t−1)).

It is also interesting to consider graphs of bounded degrees. Haxell, Szabó and Tardos proved mcc2(G) ≤ 20000 for every graph G of maximum degree 5. We show that there are n-vertex 7-regular graphs G with mcc2(G)=Ω(n), and more sharply, for every ϵ > 0 there exists cϵ > 0 and n-vertex graphs of maximum degree 7, average degree at most 6 + ϵ for all subgraphs, and with mcc2(G) ≥ cϵn. For 6-regular graphs it is known only that the maximum order of magnitude of mcc2 is between and n.

We also offer a Ramsey-theoretic perspective of the quantity mcct(G).

A famous result of Freĭman describes the sets A, of integers, for which |A+A| ≤ K|A|. In this short note we address the analogous question for subsets of vector spaces over . Specifically we show that if A is a subset of a vector space over with |A+A| ≤ K|A| then A is contained in a coset of size at most 2O(K3/2 log K)|A|, which improves upon the previous best, due to Green and Ruzsa, of 2O(K2)|A|. A simple example shows that the size may need to be at least 2Ω(K)|A|.

We obtain large-deviation approximations for the empirical distribution for a general family of occupancy problems. In the general setting, balls are allowed to fall in a given urn depending on the urn's contents prior to the throw. We discuss a parametric family of statistical models that includes Maxwell–Boltzmann, Bose–Einstein and Fermi–Dirac statistics as special cases. A process-level large-deviation analysis is conducted and the rate function for the original problem is then characterized, via the contraction principle, by the solution to a calculus of variations problem. The solution to this variational problem is shown to coincide with that of a simple finite-dimensional minimization problem. As a consequence, the large-deviation approximations and related qualitative information are available in more-or-less explicit form.

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.

We prove tail estimates for variables of the form ∑if(Xi), where (Xi)i is a sequence of states drawn from a reversible Markov chain, or, equivalently, from a random walk on an undirected graph. The estimates are in terms of the range of the function f, its variance, and the spectrum of the graph. The purpose of our estimates is to determine the number of chain/walk samples which are required for approximating the expectation of a distribution on vertices of a graph, especially an expander. The estimates must therefore provide information for fixed number of samples (as in Gillman's [4]) rather than just asymptotic information. Our proofs are more elementary than other proofs in the literature, and our results are sharper. We obtain Bernstein- and Bennett-type inequalities, as well as an inequality for sub-Gaussian variables.

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

Let the random graph Rn be drawn uniformly at random from the set of all simple planar graphs on n labelled vertices. We see that with high probability the maximum degree of Rn is Θ(ln n). We consider also the maximum size of a face and the maximum increase in the number of components on deleting a vertex. These results extend to graphs embeddable on any fixed surface.

For a fixed ρ ∈ [0, 1], what is (asymptotically) the minimal possible density g3(ρ) of triangles in a graph with edge density ρ? We completely solve this problem by proving that where is the integer such that .

Let ai,bi, i = 0, 1, 2, . . . be drawn uniformly and independently from the unit interval, and let t be a fixed real number. Let a site (i, j) ∈ be open if ai + bj ≤ t, and closed otherwise. We obtain a simple, exact expression for the probability Θ(t) that there is an infinite path (oriented or not) of open sites, containing the origin. Θ(t) is continuous and has continuous first derivative except at the critical point (t=1), near which it has critical exponent (3 − )/2.

Given a set of s points and a set of n2 lines in three-dimensional Euclidean space such that each line is incident to n points but no n lines are coplanar, we show that s = Ω(n11/4). This is the first non-trivial answer to a question recently posed by Jean Bourgain.