We provide an alternative, simpler and more general derivation of the Clarkson–Shor probabilistic technique [7] and use it to obtain several extensions and new combinatorial bounds.

A set $S\subset \R^d$ is $C$-Lipschitz in the$x_i$-coordinate, where $C>0$ is a real number, if, for every two points $a,b\in S$, we have $|a_i-b_i|\leq C \max\{|a_j-b_j|\sep j=1,2,\ldots,d,\,j\neq i\}$. Motivated by a problem of Laczkovich, the author asked whether every $n$-point set in $\Rd$ contains a subset of size at least $cn^{1-1/d}$ that is $C$-Lipschitz in one of the coordinates, for suitable constants $C$ and $c>0$ (depending on $d$). This was answered negatively by Alberti, Csörnyei and Preiss. Here it is observed that a combinatorial result of Ruzsa and Szemerédi implies the existence of a 2-Lipschitz subset of size $n^{1/2}\varphi(n)$ in every $n$-point set in $\R^3$, where $\varphi(n)\to\infty$ as $n\to\infty$.

This paper shows that the largest possible contrast $C_{k,n}$ in a $k$-out-of-$n$ secret sharing scheme is approximately $4^{-(k-1)}$. More precisely, we show that $4^{-(k-1)} \leq C_{k,n} \leq 4^{-(k-1)}n^k/(n(n-1)\cdots(n-(k-1)))$. This implies that the largest possible contrast equals $4^{-(k-1)}$ in the limit when $n$ approaches infinity. For large $n$, the above bounds leave almost no gap. For values of $n$ that come close to $k$, we will present alternative bounds (being tight for $n=k$). The proofs of our results proceed by finding a relationship between the largest possible contrast in a secret sharing scheme and the smallest possible approximation error in problems occurring in approximation theory.

We prove that, for all values of the edge probability $p(n)$, the largest eigenvalue of the random graph $G(n, p)$ satisfies almost surely $\lambda_1(G)=(1+o(1))\max\{\sqrt{\Delta}, np\}$, where Δ is the maximum degree of $G$, and the o(1) term tends to zero as $\max\{\sqrt{\Delta}, np\}$ tends to infinity.

We show that the constant colourings and the one-to-one colourings are insufficient for a canonical version of a certain theorem in Ramsey theory.

We investigate the asymptotic behaviour of the average displacement of the simple random walk on the Sierpiński graph. The existence of an oscillating factor in this asymptotics is shown rigorously. The proof depends mainly on the analysis of the corresponding generating function. Using a functional equation and techniques from complex analysis we obtain the desired properties of this generating function.

This paper is concerned with path techniques for quantitative analysis of the logarithmic Sobolev constant on a countable set. We present new upper bounds on the logarithmic Sobolev constant, which generalize those given by Sinclair [20], in the case of the spectral gap constant involving path combinatorics. Some examples of applications are given. Then, we compare our bounds to the Hardy constant in the particular case of birth and death processes. Finally, following the approach of Rosenthal in [18], we generalize our bounds to continuous sets.

We study the following one-person game against a random graph process: the Player's goal is to 2-colour a random sequence of edges of a complete graph on n vertices, avoiding a monochromatic triangle for as long as possible. The game is over when a monochromatic triangle is created. The online version of the game requires that the Player should colour each edge as it comes, before looking at the next edge.

While it is not hard to prove that the expected length of this game is about , the proof of the upper bound suggests the following relaxation: instead of colouring online, the random graph is generated in only two rounds, and the Player colours the edges of the first round before the edges of the second round are thrown in. Given the size of the first round, how many edges can there be in the second round for the Player to be likely to win? In the extreme case, when the first round consists of a random graph with edges, where c is a positive constant, we show that the Player can win with high probability only if constantly many edges are generated in the second round.

Determining the distribution of the number of empty urns after a number of balls have been thrown randomly into the urns is a classical and well understood problem. We study a generalization: Given a finite alphabet of size σ and a word length q, what is the distribution of the number X of words (of length q) that do not occur in a random text of length n+q−1 over the given alphabet? For q=1, X is the number Y of empty urns with σ urns and n balls. For q[ges ]2, X is related to the number Y of empty urns with σq urns and n balls, but the law of X is more complicated because successive words in the text overlap. We show that, perhaps surprisingly, the laws of X and Y are not as different as one might expect, but some problems remain currently open.

Suppose $n$ circular arcs of lengths $\len_i \in [0,1],0\leq i<n$, are placed uniformly at random on a unit length circle. We study the maximum overlap, i.e., the number of arcs that overlap at the same position of the circle. In particular, we give almost exact tail bounds for this random variable. By applying these tail bounds we can characterize the expected maximum overlap exactly up to constant factors in lower order terms. We illustrate the strength of our results by presenting new performance guarantees for three algorithmic applications: minimizing rotational delays for disks, scheduling accesses to parallel disks, and allocating memory blocks to limit cache interference misses.

In this paper we prove that, if a collection of sets [Ascr ] is union-closed, then the average set size of $\cal A is at least $\frac{1}{2}\log_2 (\vert{\cal A}\vert)$.

Let G be a simple 3-connected graph with at least five vertices. Tutte [13] showed that G has at least one contractible edge. Thomassen [11] gave a simple proof of this fact and showed that contractible edges have many applications. In this paper, we show that there are at most $\frac{|V(G)|}{5}$ vertices that are not incident to contractible edges in a 3-connected graph G. This bound is best-possible. We also show that if a vertex v is not incident to any contractible edge in G, then v has at least four neighbours having degree three, and each such neighbour is incident to exactly two contractible edges. We give short proofs of several results on contractible edges in 3-connected graphs as well. We also study the contractible elements for k-connected matroids. We partially solve an open problem for regular matroids.

This is a continuation of our work on quasi-random graph properties. The class of quasi-random graphs is defined by certain equivalent graph properties possessed by random graphs. One of the most important of these properties is that, for fixed $\nu$, every fixed sample graph $L_\nu$ has the same frequency in $G_n$ as in the $p$-random graph. (This holds for both induced and not necessarily induced containment.) In [9] we proved that, if the frequency of just one fixed $L_\nu$ – as a not necessarily induced subgraph – in every ‘large’ induced subgraph $F_h\subseteq G_n$ is the same as for the random graphs, then $(G_n)$ is quasi-random. Here we shall investigate the analogous problem for induced subgraphs $L_\nu$. In such cases $(G_n)$ is not necessarily quasi-random.

We shall prove, among other things, that, for every regular sample graph $L_\nu$, $\nu\geqslant 4$, if the number of induced copies of $L_\nu$ in every induced $F_h\subseteq G_n$ is asymptotically the same as in a $p$-random graph (up to an error term $o(n^\nu)$), then $(G_n)$ is the union of (at most) two quasi-random graph sequences, with possibly distinct attached probabilities (assuming that $p\in (0,1)$, $e(L_\nu)>0$, and $L_\nu\ne K_\nu$).

We conjecture the same conclusion for every $L_\nu$ with $\nu\ge 4$, i.e., even if we drop the assumption of regularity.

We shall reduce the general problem to solving a system of polynomials. This gives a ‘simple’ algorithm to decide the problem for every given $L_\nu$.

We consider analogues of van der Waerden's theorem and Szemerédi's theorem, where arithmetic progressions are replaced by binary trees with a fixed distance between successive vertices. The proofs are based on some novel recurrence properties for Markov processes.

In a previous paper we found a relation between the ranks of co-diagonal matrices (matrices with zeroes in their diagonal and nonzeroes elsewhere) and the quality of explicit Ramsey graph constructions. We also gave a construction based on the BBR polynomial of Barrington, Beigel and Rudich. In the present work we give another construction for low-rank co-diagonal matrices, based on a modular sieve formula.

The branching random walk on a regular graph turns out to be particularly easy to analyse using results for the corresponding simple random walk. In this way, one can show that there is an intermediate phase of weak survival if and only if the graph is nonamenable. No such simple analysis holds more generally, and it is known that the nonamenability equivalence does not extend to general connected graphs of bounded degree (although we observe that it does hold for such graphs if the branching random walk is modified in a certain natural way). The most important general class of (bounded degree, connected) graphs for which it is thought that the equivalence may hold is that of quasi-transitive graphs: we show that this is indeed the case.

The bond percolation critical probability of the Kagomé lattice is greater than 0.5209 and less than 0.5291. The proof of these bounds uses the substitution method, comparing the percolative behaviour of the Kagomé lattice bond model with that of the exactly solved hexagonal lattice bond model via stochastic ordering.

A finite or infinite matrix A with rational entries is called partition regular if, whenever the natural numbers are finitely coloured, there is a monochromatic vector x with . Many of the classical theorems of Ramsey Theory may naturally be interpreted as assertions that particular matrices are partition regular.

While in the finite case partition regularity is well understood, very little is known in the infinite case. Our aim in this paper is to present some of the natural and appealing open problems in the area.

Given a positive integer n and a family of graphs, let denote the maximum number of colours in an edge-colouring of such that no subgraph of belonging to has distinct colours on its edges. Erdös, Simonovits and Sós [6] conjectured for fixed k with that . This has been proved for . For general k, in this paper we improve the previous bound of to . For even k, we further improve it to . We also prove that , which is sharp.

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.