The aim of this paper is to prove a Turán-type theorem for random graphs. For $\gamma >0$ and graphs $G$ and $H$, write $G\to_\gamma H$ if any $\gamma$-proportion of the edges of $G$ spans at least one copy of $H$ in $G$. We show that for every graph $H$ and every fixed real $\delta>0$, almost every graph $G$ in the binomial random graph model $\cG(n,q)$, with $q=q(n)\gg((\log n)^4/n)^{1/d(H)}$, satisfies $G\to_{(\chi(H)-2)/(\chi(H)-1)+\delta}H$, where as usual $\chi(H)$ denotes the chromatic number of $H$ and $d(H)$ is the ‘degeneracy number’ of $H$.

Since $K_l$, the complete graph on $l$ vertices, is $l$-chromatic and $(l-1)$-degenerate, we infer that for every $l\geq2$ and every fixed real $\delta>0$, almost every graph $G$ in the binomial random graph model $\cG(n,q)$, with $q=q(n)\gg((\log n)^4/n)^{1/(l-1)}$, satisfies $G\to_{(l-2)/(l-1)+\delta}K_l$.

Baranyai's partition theorem states that the edges of the complete $r$-graph on $n$ vertices can be partitioned into $1$-factors provided that $r$ divides $n$. Fon-der-Flaass has conjectured that for $r=3$ such a partitioning exists with the property that any two $1$-factors are ‘far apart’ in some natural sense.

Our aim in this note is to prove that the Fon-der-Flaass conjecture is not always true: it fails for $n=12$. Our methods are based on some new ‘auxiliary’ hypergraphs.

Given a subset $S$ of an abelian group $G$ and an integer $k\geq 1$, the $k$-deck of $S$ is the function that assigns to every $T\subseteq G$ with at most $k$ elements the number of elements $g\in G$ with $g+T\subseteq S$. The reconstruction problem for an abelian group $G$ asks for the minimal value of $k$ such that every subset $S$ of $G$ is determined, up to translation, by its $k$-deck. This minimal value is the set-reconstruction number$r_{\rm set}(G)$ of $G$; the corresponding value for multisets is the reconstruction number$r(G)$.

Previous work had given bounds for the set-reconstruction number of cyclic groups: Alon, Caro, Krasikov and Roditty [1] showed that $r_{\rm set}({\mathbb{Z}}_n)<\log_2n$ and Radcliffe and Scott [15] that $r_{\rm set}({\mathbb{Z}}_n)<9\frac{\ln n}{\ln\ln n}$. We give a precise evaluation of $r(G)$ for all abelian groups $G$ and deduce that $r_{\rm set}({\mathbb{Z}}_n)\leq 6$.

The Lehmer–Euclid Algorithm is an improvement of the Euclid Algorithm when applied to large integers. The original Lehmer–Euclid Algorithm replaces divisions on multi-precision integers by divisions on single-precision integers. Here we study a slightly different algorithm that replaces computations on $n$-bit integers by computations on $\mu n$-bit integers. This algorithm depends on the truncation degree $\mu\in ]0, 1[$ and is denoted as the ${\mathcal{LE}}_\mu$ algorithm. The original Lehmer–Euclid Algorithm can be viewed as the limit of the ${\mathcal{LE}}_\mu$ algorithms for $\mu \to 0$. We provide here a precise analysis of the ${\mathcal{LE}}_\mu$ algorithm. For this purpose, we are led to study what we call the Interrupted Euclid Algorithm. This algorithm depends on some parameter $\alpha \in [0, 1]$ and is denoted by ${\mathcal E}_{\alpha}$. When running with an input $(a, b)$, it performs the same steps as the usual Euclid Algorithm, but it stops as soon as the current integer is smaller than $a^\alpha$, so that ${\mathcal E}_{0}$ is the classical Euclid Algorithm. We obtain a very precise analysis of the algorithm ${\mathcal E}_{\alpha}$, and describe the behaviour of main parameters (number of iterations, bit complexity) as a function of parameter $\alpha$. Since the Lehmer–Euclid Algorithm ${\mathcal {LE}}_\mu$ when running on $n$-bit integers can be viewed as a sequence of executions of the Interrupted Euclid Algorithm ${\mathcal E}_{1/2}$ on $\mu n $-bit integers, we then come back to the analysis of the ${\mathcal {LE}}_\mu$ algorithm and obtain our results.

Using representation theory, we obtain a necessary and sufficient condition for a discrete-time Markov chain on a finite state space $E$ to be representable as $$\Psi_n \Psi_{n-1} \cdots \Psi_1 z,\quad n \geq 0,$$ for any $z \in E$, where the $\Psi_i$ are independent, identically distributed random permutations taking values in some given transitive group of permutations on $E$. The condition is particularly simple when the group is 2-transitive on $E$. We also work out the explicit form of our condition for the dihedral group of symmetries of a regular polygon.

An additive decomposition of a set $I$ of nonnegative integers is an expression of $I$ as the arithmetic sum of two other such sets. If the smaller of these has $p$ elements, we have a $p$-decomposition. If $I$ is obtained by randomly removing $n^{\alpha}$ integers from $\{0,\dots,n-1\}$, decomposability translates into a balls-and-urns problem, which we start to investigate (for large $n$) by first showing that the number of $p$-decompositions exhibits a threshold phenomenon as $\alpha$ crosses a $p$-dependent critical value. We then study in detail the distribution of the number of 2-decompositions. For this last case we show that the threshold is sharp and we establish the threshold function.

Let $F\,{=}\,\{H_1,\ldots,H_k\}$ be a family of graphs. A graph $G$ is called totally$F$-decomposable if for every linear combination of the form $\alpha_1 e(H_1) \,{+}\,{\cdots}\,{+}\,\alpha_k e(H_k) \,{=}\, e(G)$ where each $\alpha_i$ is a nonnegative integer, there is a colouring of the edges of $G$ with $\alpha_1\,{+}\,{\cdots}\,{+}\,\alpha_k$ colours such that exactly $\alpha_i$ colour classes induce each a copy of $H_i$, for $i\,{=}\,1,\ldots,k$. We prove that if $F$ is any fixed nontrivial family of trees then $\log n/n$ is a sharp threshold function for the property that the random graph $G(n,p)$ is totally $F$-decomposable. In particular, if $H$ is a tree with more than one edge, then $\log n/n$ is a sharp threshold function for the property that $G(n,p)$ contains $\lfloor e(G)/e(H) \rfloor$ edge-disjoint copies of $H$.

Mader asked whether every $C_4$-free graph $G$ contains a subdivision of a complete graph whose order is at least linear in the average degree of $G$. We show that there is a subdivision of a complete graph whose order is almost linear. More generally, we prove that every $K_{s,t}$-free graph of average degree $r$ contains a subdivision of a complete graph of order $r^{\frac{1}{2}{+}\frac{1}{2(s-1)}-o(1)}$.

Given a set $L$ of $n$ lines in ${\mathbb R}^3$, joints are points in ${\mathbb R}^3$ that are incident to at least three non-coplanar lines in $L$. We show that there are at most $O(n^{5/3})$ incidences between $L$ and the set of its joints.

This result leads to related questions about incidences between $L$ and a set $P$ of $m$ points in ${\mathbb R}^3$. First, we associate with every point $p \in P$ the minimum number of planes it takes to cover all lines incident to $p$. Then the sum of these numbers is at most \[ O\big(m^{4/7}n^{5/7}+m+n\big).\] Second, if each line forms a fixed given non-zero angle with the $xy$-plane – we say the lines are equally inclined – then the number of (real) incidences is at most \[ O\big(\min\big\{m^{3/4}n^{1/2}\kappa(m),\ m^{4/7}n^{5/7}\big\} + m + n\big) , \] where $\kappa(m) \,{=}\, (\log m)^{O(\alpha^2(m))}$, and $\alpha(m)$ is the slowly growing inverse Ackermann function. These bounds are smaller than the tight Szemerédi–Trotter bound for point–line incidences in $\reals^2$, unless both bounds are linear. They are the first results of this type on incidences between points and $1$-dimensional objects in $\reals^3$. This research was stimulated by a question raised by G. Elekes.

We compute the fat-shattering function and the level fat-shattering function for important classes of affine functions. We observe that the level fat-shattering function and the fat-shattering function are identical for these classes. In addition we observe that the notion that adding the constant term to linear functions increases the dimension by at most 1 is incorrect for fat-shattering and level fat-shattering.

The random assignment problem is to minimize the cost of an assignment in an $n\times n$ matrix of random costs. In this paper we study the problem for some integer-valued cost distributions. We consider both uniform distributions on $1,2,\dots ,m$, for $m=n$ or $n^2$, and random permutations of $1,2,\dots ,n$ for each row, or of $1,2,\dots ,n^2$ for the whole matrix. We find the limit of the expected cost for the ‘$n^2$’ cases, and prove bounds for the ‘$n$’ cases. This is done by simple coupling arguments together with recent results of Aldous for the continuous case. We also present a simulation study of these cases.

I show that the zeros of the chromatic polynomials $P_G(q)$ for the generalized theta graphs $\Theta^{(s,p)}$ are, taken together, dense in the whole complex plane with the possible exception of the disc $|q-1| < 1$. The same holds for their dichromatic polynomials (alias Tutte polynomials, alias Potts-model partition functions) $Z_G(q,v)$ outside the disc $|q+v| < |v|$. An immediate corollary is that the chromatic roots of not-necessarily-planar graphs are dense in the whole complex plane. The main technical tool in the proof of these results is the Beraha–Kahane–Weiss theorem on the limit sets of zeros for certain sequences of analytic functions, for which I give a new and simpler proof.

A signing of a graph $G=(V,E)$ is a function $s:E \rightarrow \{-1,1\}$. A signing defines a graph $\widehat{G}$, called a {\em 2-lift of $G$}, with vertex set $V(G)\times\{-1,1\}$. The vertices $(u,x)$ and $(v,y)$ are adjacent iff $(u,v) \in E(G)$, and $x \cdot y = s(u,v)$. The corresponding signed adjacency matrix$A_{G,s}$ is a symmetric $\{-1,0,1\}$-matrix, with $(A_{G,s})_{u,v} = s(u,v)$ if $(u,v) \in E$, and $0$ otherwise.

This article proposes a surprisingly simple framework for the random generation of combinatorial configurations based on what we call Boltzmann models. The idea is to perform random generation of possibly complex structured objects by placing an appropriate measure spread over the whole of a combinatorial class – an object receives a probability essentially proportional to an exponential of its size. As demonstrated here, the resulting algorithms based on real-arithmetic operations often operate in linear time. They can be implemented easily, be analysed mathematically with great precision, and, when suitably tuned, tend to be very efficient in practice.

We show that for every $\varepsilon\,{>}\,0$ there exists an $r_0\,{=}\,r_0(\varepsilon)$ such that, for all integers $r\,{\ge}\, r_0$, every graph of average degree at least $r+\varepsilon$ and girth at least 1000 contains a subdivision of $K_{r+2}$. Combined with a result of Mader this implies that, for every $\varepsilon\,{>}\,0$, there exists an $f(\varepsilon)$ such that, for all $r\,{\ge}\, 2$, every graph of average degree at least $r+\varepsilon$ and girth at least $f(\varepsilon)$ contains a subdivision of $K_{r+2}$. We also prove a more general result concerning subdivisions of arbitrary graphs.

We look at a model of random graphs suggested by Gilbert: given an integer $n$ and $\delta > 0$, scatter $n$ vertices independently and uniformly on a metric space, and then add edges connecting pairs of vertices of distance less than $\delta$ apart.

We consider the asymptotics when the metric space is the interval [0, 1], and $\delta = \delta(n)$ is a function of $n$, for $n \to \infty$. We prove that every upwards closed property of (ordered) graphs has at least a weak threshold in this model on this metric space. (But we do find a metric space on which some upwards closed properties do not even have weak thresholds in this model.) We also prove that every upwards closed property with a threshold much above connectivity's threshold has a strong threshold. (But we also find a sequence of upwards closed properties with lower thresholds that are strictly weak.)

As usual, let us write $G_{n,p}$ for a random graph with vertex set $[n]=\{1, 2, \dots , n\}$, in which the edges are chosen independently, with probability $p$. Similarly, $G_{n,m}$ is a random graph on $[n]$ with $m$ edges. For $p=m/({{n}\atop{2}})$, in many respects, the random graphs $G_{n,m}$ and $G_{n,p}$ are practically indistinguishable. (See [4] for an introduction to random graphs.) When in the late 1950s and early 1960s Erdős and Rényi founded the theory of random graphs, one of the most important problems they left open was the determination of the chromatic number of a random graph $G_{n,m}$. The original question concerned the case $m=O(n)$, but when in the 1970s Erdős popularized the problem, he was asking for good estimates on the chromatic number of $G_{n,m}$ for $m \sim n^2/4$ (equivalently, the chromatic number of $G_{{n,1/2}}$).

We give a quantitative proof that, for sufficiently large $N$, every subset of $[N]^2$ of size at least $\delta N^2$ contains a square, i.e., four points with coordinates $\{(a,b),(a+d,b),(a,b+d),(a+d,b+d)\}$.

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.