The study of asymptotics of random permutations was initiated by Erdős and Turáan, in a series of papers from 1965 to 1968, and has been much studied since. Recent developments in permutation group theory make it reasonable to ask questions with a more group-theoretic flavour. Two examples considered here are membership in a proper transitive subgroup, and the intersection of a subgroup with a random conjugate. These both arise from other topics (quasigroups, bases for permutation groups, and design constructions).

Let Per f(n) denote the set of all perfect graphs on n vertices and let Berge(n) denote the set of all Berge graphs on n vertices. The strong perfect graph conjecture states that Per f(n) = Berge(n) for all n. In this paper we prove that this conjecture is at least asymptotically true, i.e. we show that

Total path length, or search cost, for a rooted tree is defined as the sum of all root-to-node distances. Let Tn be the total path length for a random recursive tree of order n. Mahmoud [10] showed that Wn := (Tn − E[Tn])/n converges almost surely and in L2 to a nondegenerate limiting random variable W. Here we give recurrence relations for the moments of Wn and of W and show that Wn converges to W in Lp for each 0 < p < ∞. We confirm the conjecture that the distribution of W is not normal. We also show that the distribution of W is characterized among all distributions having zero mean and finite variance by the distributional identity

formula here

where [Escr ](x) := − x ln x − (1 minus; x) ln(1 − x) is the binary entropy function, U is a uniform (0, 1) random variable, W* and W have the same distribution, and U, W and W* are mutually independent. Finally, we derive an approximation for the distribution of W using a Pearson curve density estimator.

By the complexity of a graph we mean the minimum number of union and intersection operations needed to obtain the whole set of its edges starting from stars. This measure of graphs is related to the circuit complexity of boolean functions.

We prove some lower bounds on the complexity of explicitly given graphs. This yields some new lower bounds for boolean functions, as well as new proofs of some known lower bounds in the graph-theoretic framework. We also formulate several combinatorial problems whose solution would have intriguing consequences in computational complexity.

For a finite multigraph G, the reliability function of G is the probability RG(q) that if each edge of G is deleted independently with probability q then the remaining edges of G induce a connected spanning subgraph of G; this is a polynomial function of q. In 1992, Brown and Colbourn conjectured that, for any connected multigraph G, if q ∈ [Copf ] is such that RG(q) = 0 then [mid ]q[mid ] [les ] 1. We verify that this conjectured property of RG(q) holds if G is a series-parallel network. The proof is by an application of the Hermite–Biehler theorem and development of a theory of higher-order interlacing for polynomials with only real nonpositive zeros. We conclude by establishing some new inequalities which are satisfied by the f-vector of any matroid without coloops, and by discussing some stronger inequalities which would follow (in the cographic case) from the Brown–Colbourn conjecture, and are hence true for cographic matroids of series-parallel networks.

We study the coefficients of algebraic functions ∑n≥0fnzn. First, we recall the too-little-known fact that these coefficients fn always admit a closed form. Then we study their asymptotics, known to be of the type fn ~ CAnnα. When the function is a power series associated to a context-free grammar, we solve a folklore conjecture: the critical exponents α cannot be 1/3 or −5/2; they in fact belong to a proper subset of the dyadic numbers. We initiate the study of the set of possible values for A. We extend what Philippe Flajolet called the Drmota–Lalley–Woods theorem (which states that α=−3/2 when the dependency graph associated to the algebraic system defining the function is strongly connected). We fully characterize the possible singular behaviours in the non-strongly connected case. As a corollary, the generating functions of certain lattice paths and planar maps are not determined by a context-free grammar (i.e., their generating functions are not ℕ-algebraic). We give examples of Gaussian limit laws (beyond the case of the Drmota–Lalley–Woods theorem), and examples of non-Gaussian limit laws. We then extend our work to systems involving non-polynomial entire functions (non-strongly connected systems, fixed points of entire functions with positive coefficients). We give several closure properties for ℕ-algebraic functions. We end by discussing a few extensions of our results (infinite systems of equations, algorithmic aspects).

In this paper we study random linear systems with $k > 3$ variables per equation over the finite field GF(2), or equivalently k-XOR-CNF formulas. In a previous paper Creignou and Daudé proved that there exists a phase transition exhibiting a sharp threshold, for the consistency (satisfiability) of such systems (formulas). The control parameter for this transition is the ratio of the number of equations to the number of variables, and the scale for which the transition occurs remains somewhat elusive. In this paper we establish, for any $k > 3$, non-trivial lower and upper estimates of the value of the control ratio for which the phase transition occurs. For $k=3$ we get 0.89 and 0.93, respectively. Moreover, we give experimental results for $k=3$ suggesting that the critical ratio is about 0.92. Our estimates are clearly close to the critical ratio.

A lattice basis bi,…,bm is called block reduced with block size β if for every β consecutive vectors bi,…,bi+β−1, the orthogonal projections of bi,…,bi+β−1 in span(bi,…,bi−1)⊥ are reduced in the sense of Hermite and Korkin–Zolotarev. Let λi denote the successive minima of lattice L, and let b1,…,bm be a basis of L that is block reduced with block size β. We prove that for i = 1,…, m

where γβ is the Hermite constant for dimension β. For block size β = 3 and odd rank m ≥ 3, we show that

where the maximum is taken over all block reduced bases of all lattices L. We present critical block reduced bases achieving this maximum. Using block reduced bases, we improve Babai's construction of a nearby lattice point. Given a block reduced basis with block size β of the lattice L, and given a point x in the span of L, a lattice point υ can be found in time βO(β) satisfying

These results also give improvements on the method of solving integer programming problems via basis reduction.

Let ${\cal H}$ be a 3-uniform hypergraph on an $n$-element vertex set $V$. The neighbourhood of $a,b\in V$ is $N(ab):= \{x: abx\in E({\cal H})\} $. Such a 3-graph has independent neighbourhoods if no $N(ab)$ contains an edge of ${\cal H}$. This is equivalent to ${\cal H}$ not containing a copy of $\mathbb{F} :=\{ abx$, $aby$, $abz$, $xyz\}$.

In this paper we prove an analogue of the Andrásfai–Erdös–Sós theorem for triangle-free graphs with minimum degree exceeding $2n/5$. It is shown that any $\mathbb{F}$-free 3-graph with minimum degree exceeding $(\frac{4}{9}-\frac{1}{125})\binom{n}{2}$ is bipartite, (for $n> n_0$), i.e., the vertices of ${\cal H}$ can be split into two parts so that every triple meets both parts.

This is, in fact, a Turán-type result. It solves a problem of Erdös and T.Sós, and answers a question of Mubayi and Rödl that

\[\ex(n,\mathbb{F}_{3,2})=\max\limits_{\alpha}(n-\alpha)\left({\alpha\atop2}\right)\].

Here the right-hand side is $\frac{4}{9}\binom{n}{3}+O(n^2)$. Moreover $e({\cal H})={\rm ex}(n,\mathbb{F})$ is possible only if $V({\cal H})$ can be partitioned into two sets $A$ and $B$ so that each triple of ${\cal H}$ intersects $A$ in exactly two vertices and $B$ in one.

We introduce the notion of a stable instance for a discrete optimization problem, and argue that in many practical situations only sufficiently stable instances are of interest. The question then arises whether stable instances of NP-hard problems are easier to solve, and in particular, whether there exist algorithms that solve in polynomial time all sufficiently stable instances of some NP-hard problem. The paper focuses on the Max-Cut problem, for which we show that this is indeed the case.

We are given a graph G with n vertices, where a random subset of k vertices has been made into a clique, and the remaining edges are chosen independently with probability $\frac12$. This random graph model is denoted $G(n,\frac12,k)$. The hidden clique problem is to design an algorithm that finds the k-clique in polynomial time with high probability. An algorithm due to Alon, Krivelevich and Sudakov [3] uses spectral techniques to find the hidden clique with high probability when $k = c \sqrt{n}$ for a sufficiently large constant c > 0. Recently, an algorithm that solves the same problem was proposed by Feige and Ron [12]. It has the advantages of being simpler and more intuitive, and of an improved running time of O(n2). However, the analysis in [12] gives a success probability of only 2/3. In this paper we present a new algorithm for finding hidden cliques that both runs in time O(n2) (that is, linear in the size of the input) and has a failure probability that tends to 0 as n tends to ∞. We develop this algorithm in the more general setting where the clique is replaced by a dense random graph.

For a graph G, let f(G) denote the maximum number of edges in a cut of G. For an integer m and for a fixed graph H, let $f(m,H)$ denote the minimum possible cardinality of $f(G)$, as G ranges over all graphs on m edges that contain no copy of H. In this paper we study this function for various graphs H. In particular we show that for any graph H obtained by connecting a single vertex to all vertices of a fixed nontrivial forest, there is a $c(H) >0$ such that $f(m,H) \geq \frac{m}{2} + c(H) m^{4/5}$, and that this is tight up to the value of $c(H)$. We also prove that for any even cycle $C_{2k}$ there is a $c(k)>0$ such that $f(m,C_{2k}) \geq \frac{m}{2} + c(k) m^{(2k+1)/(2k+2)}$, and that this is tight, up to the value of $c(k)$, for $2k\in \{4,6,10\}$. The proofs combine combinatorial, probabilistic and spectral techniques.

The square lattice site percolation model critical probability is shown to be at most .679492, improving the best previous mathematically rigorous upper bound. This bound is derived by extending the substitution method to apply to site percolation models.

The conditional independences within a system of four discrete random variables are studied simultaneously. The problem of where independences can occur at the same time, called the problem of probabilistic representability, is attacked by an analysis of cones of polymatroids. New results on the cone of all polymatroids satisfying Ingleton inequalities imply a substantial reduction of the problem and an explicit description of the remaining open cases.†

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 $\{(a,b),(a+d,b),(a,b+d),(a+d,b+d)\}$.

Let ⊂ 2[n] be a family of subsets of {1, 2,. . ., n}. For any poset H, we say is H-free if does not contain any subposet isomorphic to H. Katona and others have investigated the behaviour of La(n, H), which denotes the maximum size of H-free families ⊂ 2[n]. Here we use a new approach, which is to apply methods from extremal graph theory and probability theory to identify new classes of posets H, for which La(n, H) can be determined asymptotically as n → ∞ for various posets H, including two-end-forks, up-down trees, and cycles C4k on two levels.

Let $C$ be a code of length $n$ over an alphabet of $q$ letters. A codeword $y$ is called a descendant of a set of $t$ codewords $\{x^1,\dots,x^t\}$ if $y_i \in \{x^1_i,\dots,x^t_i\}$ for all $i=1,\dots,n$. A code is said to have the Identifiable Parent Property of order $t$ if, for any word of length $n$ that is a descendant of at most $t$ codewords (parents), it is possible to identify at least one of them. Let $f_t(n,q)$ be the maximum possible cardinality of such a code. We prove that for any $t,n,q$, $(c_1(t)q)^{\frac{n}{s(t)}} < f_t(n,q) < c_2(t)q^{\lceil{\frac{n}{s(t)}}\rceil}$ where $s(t) = \lfloor(\frac{t}{2}+1)^2 \rfloor -1$ and $c_1(t),c_2(t)$ are some functions of $t$. We also show some bounds and constructions for $f_3(5,q)$, $f_3(6,q)$, and $f_t(n,q)$ when $n < s(t)$.

A family of subsets of {1, . . ., n} is called a j-junta if there exists J ⊆ {1, . . ., n}, with |J| = j, such that the membership of a set S in depends only on S ∩ J.

In this paper we provide a simple description of intersecting families of sets. Let n and k be positive integers with k < n/2, and let be a family of pairwise intersecting subsets of {1, . . ., n}, all of size k. We show that such a family is essentially contained in a j-junta , where j does not depend on n but only on the ratio k/n and on the interpretation of ‘essentially’.

When k = o(n) we prove that every intersecting family of k-sets is almost contained in a dictatorship, a 1-junta (which by the Erdős–Ko–Rado theorem is a maximal intersecting family): for any such intersecting family there exists an element i ∈ {1, . . ., n} such that the number of sets in that do not contain i is of order (which is approximately times the size of a maximal intersecting family).

Our methods combine traditional combinatorics with results stemming from the theory of Boolean functions and discrete Fourier analysis.

We describe the C2k+1-free graphs on n vertices with maximum number of edges. The extremal graphs are unique for n ∉ {3k − 1, 3k, 4k − 2, 4k − 1}. The value of ex(n, C2k+1) can be read out from the works of Bondy [3], Woodall [14], and Bollobás [1], but here we give a new streamlined proof. The complete determination of the extremal graphs is also new.

We obtain that the bound for n0(C2k+1) is 4k in the classical theorem of Simonovits, from which the unique extremal graph is the bipartite Turán graph.

We prove χ′s(G) ≤ 1.93 Δ(G)2 for graphs of sufficiently large maximum degree where χ′s(G) is the strong chromatic index of G. This improves an old bound of Molloy and Reed. As a by-product, we present a Talagrand-type inequality where we are allowed to exclude unlikely bad outcomes that would otherwise render the inequality unusable.