Erdős asked in 1962 about the value of f(n,k,l), the minimum number of k-cliques in a graph with order n and independence number less than l. The case (k,l)=(3,3) was solved by Lorden. Here we solve the problem (for all large n) for (3,l) with 4 ≤ l ≤ 7 and (k,3) with 4 ≤ k ≤ 7. Independently, Das, Huang, Ma, Naves and Sudakov resolved the cases (k,l)=(3,4) and (4,3).

A cladogram is a tree with labelled leaves and unlabelled degree-3 branchpoints. A certain Markov chain on the set of n-leaf cladograms consists of removing a random leaf (and its incident edge) and re-attaching it to a random edge. We show that the mixing time (time to approach the uniform stationary distribution) for this chain is at least O(n2) and at most O(n3).

In this paper, we study a statistical property of classes of real-valued functions that we call approximation from interpolated examples. We derive a characterization of function classes that have this property, in terms of their ‘fat-shattering function’, a notion that has proved useful in computational learning theory. The property is central to a problem of learning real-valued functions from random examples in which we require satisfactory performance from every algorithm that returns a function which approximately interpolates the training examples.

There is a close relationship between biased graph games and random graph processes. In this paper, we develop the analogy and give further interesting instances.

We propose an approach to analysing the asymptotic behaviour of Pólya urns based on the contraction method. For this, a new combinatorial discrete-time embedding of the evolution of the urn into random rooted trees is developed. A decomposition of these trees leads to a system of recursive distributional equations which capture the distributions of the numbers of balls of each colour. Ideas from the contraction method are used to study such systems of recursive distributional equations asymptotically. We apply our approach to a couple of concrete Pólya urns that lead to limit laws with normal limit distributions, with non-normal limit distributions and with asymptotic periodic distributional behaviour.

We study the number of comparisons in Hoare's Find algorithm. Using trivariate generating functions, we get an explicit expression for the variance of the number of comparisons, if we search for the jth element in a random permutation of n elements. The variance is also asymptotically evaluated under the assumption that j is proportional to n. Similar results for the number of passes (recursive calls) are given, too.

Let 3 ≤ k < n/2. We prove the analogue of the Erdős–Ko–Rado theorem for the random k-uniform hypergraph Gk(n, p) when k < (n/2)1/3; that is, we show that with probability tending to 1 as n → ∞, the maximum size of an intersecting subfamily of Gk(n, p) is the size of a maximum trivial family. The analogue of the Erdős–Ko–Rado theorem does not hold for all p when k ≫ n1/3.

We give quite precise results for k < n1/2−ϵ. For larger k we show that the random Erdős–Ko–Rado theorem holds as long as p is not too small, and fails to hold for a wide range of smaller values of p. Along the way, we prove that every non-trivial intersecting k-uniform hypergraph can be covered by k2 − k + 1 pairs, which is sharp as evidenced by projective planes. This improves upon a result of Sanders [7]. Several open questions remain.

We determine lower bounds for the number of random binary vectors, chosen uniformly from vectors of weight k, needed to obtain a dependent set.

We study the diameter of 1, the largest component of the Erdős–Rényi random graph (n, p) in the emerging supercritical phase, i.e., for p = where ε3n → ∞ and ε = o(1). This parameter was extensively studied for fixed ε > 0, yet results for ε = o(1) outside the critical window were only obtained very recently. Prior to this work, Riordan and Wormald gave precise estimates on the diameter; however, these did not cover the entire supercritical regime (namely, when ε3n → ∞ arbitrarily slowly). Łuczak and Seierstad estimated its order throughout this regime, yet their upper and lower bounds differed by a factor of .

We show that throughout the emerging supercritical phase, i.e., for any ε = o(1) with ε3n → ∞, the diameter of 1 is with high probability asymptotic to D(ε, n) = (3/ε)log(ε3n). This constitutes the first proof of the asymptotics of the diameter valid throughout this phase. The proof relies on a recent structure result for the supercritical giant component, which reduces the problem of estimating distances between its vertices to the study of passage times in first-passage percolation. The main advantage of our method is its flexibility. It also implies that in the emerging supercritical phase the diameter of the 2-core of 1 is w.h.p. asymptotic to , and the maximal distance in 1 between any pair of kernel vertices is w.h.p. asymptotic to .

We analyse a class of randomized Least Recently Used (LRU) cache replacement algorithms under the independent reference model with generalized Zipf's law request probabilities. The randomization was recently proposed for Web caching as a mechanism that discriminates between different document sizes. In particular, the cache maintains an ordered list of documents in the following way. When a document of size $s$ is requested and found in the cache, then with probability $p_s$ it is moved to the front of the cache; otherwise the cache stays unchanged. Similarly, if the requested document of size $s$ is not found in the cache, the algorithm places it with probability $p_s$ to the front of the cache or leaves the cache unchanged with the complementary probability $(1-p_s)$. The successive randomized decisions are independent and the corresponding success probabilities $p_s$ are completely determined by the size of the currently requested document. In the case of a replacement, the necessary number of documents that are least recently moved to the front of the cache are removed in order to accommodate the newly placed document.

In this framework, we provide explicit asymptotic characterization of the cache fault probability. Using the derived result we prove that the asymptotic performance of this class of algorithms is optimized when the randomization probabilities are chosen to be inversely proportional to document sizes. In addition, for this optimized and easy-to-implement policy, we show that its performance is within a constant factor from the optimal static algorithm.

We show that for every sufficiently large n, the number of monotone subsequences of length four in a permutation on n points is at least

\begin{equation*}\binom{\lfloor{n/3}\rfloor}{4} + \binom{\lfloor{(n+1)/3}\rfloor}{4} + \binom{\lfloor{(n+2)/3}\rfloor}{4}.\end{equation*}

A 1993 result of Alon and Füredi gives a sharp upper bound on the number of zeros of a multivariate polynomial over an integral domain in a finite grid, in terms of the degree of the polynomial. This result was recently generalized to polynomials over an arbitrary commutative ring, assuming a certain ‘Condition (D)’ on the grid which holds vacuously when the ring is a domain. In the first half of this paper we give a further generalized Alon–Füredi theorem which provides a sharp upper bound when the degrees of the polynomial in each variable are also taken into account. This yields in particular a new proof of Alon–Füredi. We then discuss the relationship between Alon–Füredi and results of DeMillo–Lipton, Schwartz and Zippel. A direct coding theoretic interpretation of Alon–Füredi theorem and its generalization in terms of Reed–Muller-type affine variety codes is shown, which gives us the minimum Hamming distance of these codes. Then we apply the Alon–Füredi theorem to quickly recover – and sometimes strengthen – old and new results in finite geometry, including the Jamison–Brouwer–Schrijver bound on affine blocking sets. We end with a discussion of multiplicity enhancements.

We study the expected value of the length Ln of the minimum spanning tree of the complete graph Kn when each edge e is given an independent uniform [0, 1] edge weight. We sharpen the result of Frieze [6] that limn→∞$\mathbb{E}$(Ln) = ζ(3) and show that

$$ \mathbb{E}(L_n)=\zeta(3)+\frac{c_1}{n}+\frac{c_2+o(1)}{n^{4/3}}, $$

We provide an alternate proof of the central limit theorem for the uctuations of the size of the giant component in sparse random graphs. In contrast with previous proofs, the argument investigates a depth-first search algorithm, through first-passage analysis using couplings and martingale limit theorems. The analysis of the first passage limiting distribution for sequences of Markov chains might be interesting in its own right. This proof naturally provides an upper bound for the rate of convergence.

The aim of this paper is to prove an inequality between relative entropy and the sum of average conditional relative entropies of the following form: for a fixed probability measure q on , ( is a finite set), and any probability measure on , (*)

$$D(p||q){\rm{\le}}C \cdot \sum\limits_{i = 1}^n {{\rm{\mathbb{E}}}_p D(p_i ( \cdot |Y_1 ,{\rm{ }}...,{\rm{ }}Y_{i - 1} ,{\rm{ }}Y_{i + 1} ,...,{\rm{ }}Y_n )||q_i ( \cdot |Y_1 ,{\rm{ }}...,{\rm{ }}Y_{i - 1} ,{\rm{ }}Y_{i + 1} ,{\rm{ }}...,{\rm{ }}Y_n )),} $$

The inequality (*) ismeaningful in product spaces, in both the discrete and the continuous case, and can be used to prove a logarithmic Sobolev inequality for q, provided uniform logarithmic Sobolev inequalities are available for qi(· |x1, …, xi−1, xi+1, …, xn), for all fixed i and fixed (x1, …, xi−1, xi+1, …, xn). Inequality (*) directly implies that the Gibbs sampler associated with q is a contraction for relative entropy.

In this paper we derive inequality (*), and thereby a logarithmic Sobolev inequality, in discrete product spaces, by proving inequalities for an appropriate Wasserstein-like distance.

The van der Waerden theorem in Ramsey theory states that, for every k and t and sufficiently large N, every k-colouring of [N] contains a monochromatic arithmetic progression of length t. Motivated by this result, Radoičić conjectured that every equinumerous 3-colouring of [3n] contains a 3-term rainbow arithmetic progression, i.e., an arithmetic progression whose terms are coloured with distinct colours. In this paper, we prove that every 3-colouring of the set of natural numbers for which each colour class has density more than 1/6, contains a 3-term rainbow arithmetic progression. We also prove similar results for colourings of . Finally, we give a general perspective on other anti-Ramsey-type problems that can be considered.

A perfect Kt-matching in a graph G is a spanning subgraph consisting of vertex-disjoint copies of Kt. A classic theorem of Hajnal and Szemerédi states that if G is a graph of order n with minimum degree δ(G) ≥ (t − 1)n/t and t|n, then G contains a perfect Kt-matching. Let G be a t-partite graph with vertex classes V1, …, Vt each of size n. We show that, for any γ > 0, if every vertex x ∈ Vi is joined to at least $\bigl ((t-1)/t + \gamma \bigr )n$ vertices of Vj for each j ≠ i, then G contains a perfect Kt-matching, provided n is large enough. Thus, we verify a conjecture of Fischer [6] asymptotically. Furthermore, we consider a generalization to hypergraphs in terms of the codegree.

We give a combinatorial proof of the result of Kahn, Kalai and Linial [16], which states that every balanced boolean function on the n-dimensional boolean cube has a variable with influence of at least . The methods of the proof are then used to recover additional isoperimetric results for the cube, with improved constants.

We also state some conjectures about optimal constants.

We study the number of random records in a binary search tree with n vertices (or equivalently, the number of cuttings required to eliminate the tree). We show that a classical limit theorem for convergence of sums of triangular arrays to infinitely divisible distributions can be used to determine the distribution of this number. The asymptotic distribution of the (normalized) number of records or cuts is found to be weakly 1-stable.

In this paper, we combine two previous works, the first being by the first author and K. Nelander, and the second by J. van den Berg and the second author, to show (1) that one can carry out a Propp–Wilson exact simulation for all Markov random fields on ℤd satisfying a certain high noise assumption, and (2) that all such random fields are a finitary image of a finite state i.i.d. process. (2) is a strengthening of the previously known fact that such random fields are so-called Bernoulli shifts.