We study the fluctuations, in the large deviations regime, of the longest increasing sub-sequence of a random i.i.d. sample on the unit square. In particular, our results yield the precise upper and lower exponential tails for the length of the longest increasing subsequence of a random permutation.

Consider the minimal weights of paths between two points in a complete graph Kn with random weights on the edges, the weights being, for instance, uniformly distributed. It is shown that, asymptotically, this is log n/n for two given points, that the maximum if one point is fixed and the other varies is 2 log n/n, and that the maximum over all pairs of points is 3 log n/n.

Some further related results are given as well, including results on asymptotic distributions and moments, and on the number of edges in the minimal weight paths.

Our main topic is the number of subsets of [1, n] which are maximal with respect to some condition such as being sum-free, having no number dividing another, etc. We also investigate some related questions.

Each edge of the standard rooted binary tree is equipped with a random weight; weights are independent and identically distibuted. The value of a vertex is the sum of the weights on the path from the root to the vertex. We wish to search the tree to find a vertex of large weight. A very natural conjecture of Aldous states that, in the sense of stochastic domination, an obvious greedy algorithm is best possible. We show that this conjecture is false. We prove, however, that in a weaker sense there is no significantly better algorithm.

We introduce the path resistance method for lower bounds on the smallest nontrivial eigenvalue of the Laplacian matrix of a graph. The method is based on viewing the graph in terms of electrical circuits: it uses clique embeddings to produce lower bounds on λ2 and star embeddings to produce lower bounds on the smallest Rayleigh quotient when there is a zero Dirichlet boundary condition. The method assigns priorities to the paths in the embedding; we show that, for an unweighted tree T, using uniform priorities for a clique embedding produces a lower bound on λ2 that is off by at most an O(log diameter(T)) factor. We show that the best bounds this method can produce for clique embeddings are the same as for a related method that uses clique embeddings and edge lengths to produce bounds.

We define uniformly spread sets as point sets in d-dimensional Euclidean space that are wobbling equivalent to the standard lattice ℤd. A linear image ϕ(ℤd) of ℤd is shown to be uniformly spread if and only if det(ϕ) = 1. Explicit geometrical and number-theoretical constructions are given. In 2-dimensional Euclidean space we obtain bounds for the wobbling distance for rotations, shearings and stretchings that are close to optimal. Our methods also allow us to analyse the discrepancy of certain billiards. Finally, we take a look at paradoxical situations and exhibit recursive point sets that are wobbling equivalent, but not recursively so.

We provide sufficient conditions for packing two hypergraphs. The emphasis is on the asymptotic case when one of the hypergraphs has a bounded degree and the other is dense. As an application, we give an alternative proof for the bipartite case of the recently developed Blow-up Lemma [12].

A family of k sets is called a Δ-system if any two sets have the same intersection. Denote by f(r, k) the least integer so that any r-uniform family of f(r, k) sets contains a Δ-system consisting of k sets. We prove that, for every fixed r, f(r, k) = kr + o(kr). Using a recent result of Molloy and Reed [5], a bound on the error term is provided for sufficiently large k.

Assemblies are decomposable combinatorial objects characterized by a sequence mi that counts the number of possible components of size i. Permutations on n elements, mappings from a set containing n elements into itself, 2-regular graphs on n vertices, and set partitions on a set of size n are all assemblies with natural decompositions. Logarithmic assemblies are characterized by constants θ > 0 and κ0 > 0 such that miκi0/ (i−1)! → θ. Random mappings, permutations and 2-regular graphs are all logarithmic assemblies, but set partitions are not.

Given a logarithmic assembly, all representatives having total size n are chosen uniformly and a component counting process C(n) = (C1(n), C2(n), …, Cn(n)) is defined, where Ci(n) is the number of components of size i. Our results also apply to C(n) distributed as the Ewens sampling formula with parameter θ. Denote the component counting process up to size at most b by Cb(n) = (C1(n), C2(n), …, Cb(n)). It is natural to approximate Cb by Zb = (Z1, Z2, …, Zb), the b-dimensional process of independent Poisson variables Zi for which the ith variable has expectation [ ]Zi = miκi0 exp((1−θ)i/n)/i!. We find asymptotics for the total variation distance between Cb(n) and Zb.

Let A be a subset of an abelian group G. The subset sum of A is the set [sum ](A) = {[sum ]x∈T[mid ]T⊂A}. We prove the following result. Let S be a generating subset of an abelian group G such that 0∉S and 14[les ][mid ]S[mid ]. Then one of the following conditions holds.

(i) [mid ][sum ](S)[mid ][ges ]min([mid ]G[mid ] −3, 3[mid ]S[mid ]−3).

(ii) There is an x∈S such that S[setmn ]{x} generates a proper subgroup of order less than (3[mid ]S[mid ]−3)/2.

As a consequence, we obtain the following open case of an old conjecture of Diderrich. Let q be a composite odd number and let G be an abelian group of order 3q. Let S be a subset of G with cardinality q+1. Then every element of G is the sum of some subset of S.

We present a practical algorithm for generating random regular graphs. For all d growing as a small power of n, the d-regular graphs on n vertices are generated approximately uniformly at random, in the sense that all d-regular graphs on n vertices have in the limit the same probability as n → ∞. The expected runtime for these ds is [Oscr ](nd2).

The problem of probabilistic representability of semimatroids over a four-element set is solved. In this problem, one looks for all combinations of conditional independences within four random variables which can occur simultaneously. New properties of the stochastic conditional independences are deduced from conditional information inequalities. Examples of four-tuples of random variables are presented to show the probabilistic representability of three non-Ingleton semimatroids.

In [1], recently published in Combinatorics, Probability and Computing, there was a typographical error. On p. 392, part (5) of Lemma 3.1, the formula should read as follows:

formula here

References

[1] Hadjicostas, P. (1998) The asymptotic proportion of subdivisions of a 2 × 2 table that result in Simpson's Paradox. Combinatorics, Probability and Computing7 387–396.

A new model of random graphs – random intersection graphs – is introduced. In this model, vertices are assigned random subsets of a given set. Two vertices are adjacent provided their assigned sets intersect. We explore the evolution of random intersection graphs by studying thresholds for the appearance and disappearance of small induced subgraphs. An application to gate matrix circuit design is presented.

We present some nice properties of the classical construction of triangle-free graphs with high chromatic number given by Blanche Descartes and its modifications. In particular, we construct colour-critical graphs and hypergraphs of high girth with moderate average degree.

A set A is called universal sum-free if, for every finite 0–1 sequence χ = (e1, …, en), either

(i) there exist i, j, where 1[les ]j<i[les ]n, such that ei = ej = 1 and i − j∈A, or

(ii) there exists t∈N such that, for 1[les ]i[les ]n, we have t + i∈A if and only if ei = 1.

It is proved that the density of each universal sum-free set is zero, which settles a problem of Cameron.

Using differential equations, we examine the GREEDY algorithm studied by Azar, Broder, Karlin and Upfal for distributed load balancing [1]. This approach yields accurate estimates of the actual load distribution, provides insight into the exponential improvement GREEDY offers over simple random selection, and allows one to prove tight concentration theorems about the loads in a straightforward manner.

Extremal graph theory has a great number of conjectures concerning the embedding of large sparse graphs into dense graphs. Szemerédi's Regularity Lemma is a valuable tool in finding embeddings of small graphs. The Blow-up Lemma, proved recently by Komlós, Sárközy and Szemerédi, can be applied to obtain approximate versions of many of the embedding conjectures. In this paper we review recent developments in the area.

Let B = [b1, …, bn] (with column vectors bi) be a basis of ℝn. Then L = [sum ]biℤ is a lattice in ℝn and A = B[top ]B is the Gram matrix of B. The reciprocal lattice L* of L has basis B* = (B−1)[top ] with Gram matrix A−1. For any nonsingular matrix A = (ai,j) with inverse A−1 = (a*i,j), let τ(A) = max1[les ]i[les ]n {[sum ]nj =1[mid ]ai,j ·a*j,j[mid ]}. Then τ(A), τ(A−1)[ges ]1 holds, with equality for an orthogonal basis. We will show that for any lattice L there is a basis with Gram matrix A such that τ(A), τ(A−1) = exp (O((ln n)2)). This generalizes a result in [8] and [20].

For any basis transformation A→Ā with Ā = T[top ]AT, T = (ti,j)∈SLn(ℤ), we will show [mid ]ti,j[mid ][les ]τ(A−1) ·τ(Ā). This implies that every integral matrix representation of a finite group is equivalent to a representation where the coefficients of the matrices representing group elements are bounded by exp (O((ln n)2)). This new bound is considerably smaller than the known (exponential) bounds for automorphisms of Minkowski-reduced lattice bases: see, for example, [6].

The quantities τ(A), τ(A−1) have the following geometric interpretation. Let V(L) [ratio ]= {x∈ℝn[mid ]∀λ∈L [ratio ][mid ]x[mid ][les ][mid ]x−λ[mid ]} be the Voronoi cell (also called the Dirichlet region) of the lattice L. For a basis B of L, we call C(B) = {[sum ]xibi, [mid ]xi[mid ][les ]1/2} the basis cell of B. Both cells define a lattice tiling of ℝn (see [6]); they coincide for an orthogonal basis. For a general basis B of L with Gram matrix A we will show V(L)[les ]τ(A−1)·C(B) and C(B)[les ]n·τ(A)·V(L).

Let [Fscr ] be a family of forbidden k-hypergraphs (k-uniform set systems). An [Fscr ]-saturated hypergraph is a maximal k-uniform set system not containing any member of [Fscr ]. As the main result we prove that, for any finite family [Fscr ], the minimum number of edges of an [Fscr ]-saturated hypergraph is O(nk−1). In particular, this implies a conjecture of Tuza. Some other related results are presented.