In a paper published in this journal, Alon, Kohayakawa, Mauduit, Moreira and Rödl proved that the minimal possible value of the normality measure of an N-element binary sequence satisfies

\begin{equation*}\biggl( \frac{1}{2} + o(1) \biggr) \log_2 N \leq \min_{E_N \in \{0,1\}^N} \mathcal{N}(E_N) \leq 3 N^{1/3} (\log N)^{2/3}\end{equation*}

In this paper we prove the decidability of the HD0L ultimate periodicity problem.

Most of the constructions of infinite words having polynomial subword complexity are quite complicated, e.g., sequences of Toeplitz, sequences defined by billiards in the cube, etc. In this paper, we describe a simple method for constructing infinite words w over a binary alphabet  { a,b }  with polynomial subword complexity pw. Assuming w contains an infinite number of a’s, our method is based on the gap function which gives the distances between consecutive b’s. It is known that if the gap function is injective, we can obtain at most quadratic subword complexity, and if the gap function is blockwise injective, we can obtain at most cubic subword complexity. Here, we construct infinite binary words w such that pw(n) = Θ(nβ) for any real number β > 1.

We prove results about the Lp-almost-periodicity of convolutions. One of these follows from a simple but rather general lemma about approximating a sum of functions in Lp, and gives a very short proof of a theorem of Green that if A and B are subsets of {1,. . .,N} of sizes αN and βN then A+B contains an arithmetic progression of length at least

\begin{equation}\exp ( c (\alpha \beta \log N)^{1/2} - \log\log N).\end{equation}

\begin{equation}\exp\biggl( c \biggl(\frac{\alpha \log N}{\log^3 2\beta^{-1}} \biggr)^{1/2} - \log( \beta^{-1} \log N) \biggr).\end{equation}

The cutwidth is an important graph-invariant in circuit layout designs. The cutwidth of agraph G is the minimum value of the maximum number of overlap edges whenG is embedded into a line. A caterpillar is a tree which yields a pathwhen all its leaves are removed. An iterated caterpillar is a tree which yields acaterpillar when all its leaves are removed. In this paper we present an exact formula forthe cutwidth of the iterated caterpillars.

Pursuit-evasion games, such as the game of Revolutionaries and Spies, are a simplified model for network security. In the game we consider in this paper, a team of r revolutionaries tries to hold an unguarded meeting consisting of m revolutionaries. A team of s spies wants to prevent this forever. For given r and m, the minimum number of spies required to win on a graph G is the spy number σ(G,r,m). We present asymptotic results for the game played on random graphs G(n,p) for a large range of p = p(n), r=r(n), and m=m(n). The behaviour of the spy number is analysed completely for dense graphs (that is, graphs with average degree at least n1/2+ε for some ε > 0). For sparser graphs, some bounds are provided.

A permutation σ describing the relative orders of the first n iterates of a point x under a self-map f of the interval I=[0,1] is called an order pattern. For fixed f and n, measuring the points x ∈ I (according to Lebesgue measure) that generate the order pattern σ gives a probability distribution μn(f) on the set of length n permutations. We study the distributions that arise this way for various classes of functions f.

Our main results treat the class of measure-preserving functions. We obtain an exact description of the set of realizable distributions in this case: for each n this set is a union of open faces of the polytope of flows on a certain digraph, and a simple combinatorial criterion determines which faces are included. We also show that for general f, apart from an obvious compatibility condition, there is no restriction on the sequence {μn(f)}n=1,2,. . ..

In addition, we give a necessary condition for f to have finite exclusion type, that is, for there to be finitely many order patterns that generate all order patterns not realized by f. Using entropy we show that if f is piecewise continuous, piecewise monotone, and either ergodic or with points of arbitrarily high period, then f cannot have finite exclusion type. This generalizes results of S. Elizalde.

For a positive integer r ≥ 2, a Kr-factor of a graph is a collection vertex-disjoint copies of Kr which covers all the vertices of the given graph. The celebrated theorem of Hajnal and Szemerédi asserts that every graph on n vertices with minimum degree at least $(1-\frac{1}{r})n contains a Kr-factor. In this note, we propose investigating the relation between minimum degree and existence of perfect Kr-packing for edge-weighted graphs. The main question we study is the following. Suppose that a positive integer r ≥ 2 and a real t ∈ [0, 1] is given. What is the minimum weighted degree of Kn that guarantees the existence of a Kr-factor such that every factor has total edge weight at least $$t\binom{r}{2}$?$ We provide some lower and upper bounds and make a conjecture on the asymptotics of the threshold as n goes to infinity.

We consider partitions of the positive integer n whose parts satisfy the following condition. For a given sequence of non-negative numbers {bk}k≥1, a part of size k appears in exactly bk possible types. Assuming that a weighted partition is selected uniformly at random from the set of all such partitions, we study the asymptotic behaviour of the largest part Xn. Let D(s)=∑k=1∞bkk−s, s=σ+iy, be the Dirichlet generating series of the weights bk. Under certain fairly general assumptions, Meinardus (1954) obtained the asymptotic of the total number of such partitions as n→∞. Using the Meinardus scheme of conditions, we prove that Xn, appropriately normalized, converges weakly to a random variable having Gumbel distribution (i.e., its distribution function equals e−e−t, −∞<t<∞). This limit theorem extends some known results on particular types of partitions and on the Bose–Einstein model of ideal gas.

A partial Steiner (n,r,l)-system is an r-uniform hypergraph on n vertices in which every set of l vertices is contained in at most one edge. A partial Steiner (n,r,l)-system is complete if every set of l vertices is contained in exactly one edge. In a hypergraph , the independence number α() denotes the maximum size of a set of vertices in containing no edge. In this article we prove the following. Given integers r,l such that r ≥ 2l − 1 ≥ 3, we prove that there exists a partial Steiner (n,r,l)-system such that

$$\alpha(\HH) \lesssim \biggl(\frac{l-1}{r-1}(r)_l\biggr)^{\frac{1}{r-1}}n^{\frac{r-l}{r-1}} (\log n)^{\frac{1}{r-1}} \quad \mbox{ as }n \rightarrow \infty.$$

In this paper we study the maximum displacement for linear probing hashing. We use the standard probabilistic model together with the insertion policy known as First-Come-(First-Served). The results are of asymptotic nature and focus on dense hash tables. That is, the number of occupied cells n and the size of the hash table m tend to infinity with ratio n/m → 1. We present distributions and moments for the size of the maximum displacement, as well as for the number of items with displacement larger than some critical value. This is done via process convergence of the (appropriately normalized) length of the largest block of consecutive occupied cells, when the total number of occupied cells n varies.

We show that any n-vertex complete graph with edges coloured with three colours contains a set of at most four vertices such that the number of the neighbours of these vertices in one of the colours is at least 2n/3. The previous best value, proved by Erdős, Faudree, Gould, Gyárfás, Rousseau and Schelp in 1989, is 22. It is conjectured that three vertices suffice.

This paper is the first application of the compensation approach (a well-established theory in probability theory) to counting problems. We discuss how this method can be applied to a general class of walks in the quarter plane +2 with a step set that is a subset of

\[ \{(-1,1),(-1,0),(-1,-1),(0,-1),(1,-1)\}\]

The analysis of randomized search heuristics on classes of functions is fundamental to the understanding of the underlying stochastic process and the development of suitable proof techniques. Recently, remarkable progress has been made in bounding the expected optimization time of a simple evolutionary algorithm, called (1+1) EA, on the class of linear functions. We improve the previously best known bound in this setting from (1.39+o(1))en ln n to en ln n+O(n) in expectation and with high probability, which is tight up to lower-order terms. Moreover, upper and lower bounds for arbitrary mutation probabilities p are derived, which imply expected polynomial optimization time as long as p = O((ln n)/n) and p = Ω(n−C) for a constant C > 0, and which are tight if p = c/n for a constant c > 0. As a consequence, the standard mutation probability p = 1/n is optimal for all linear functions, and the (1+1) EA is found to be an optimal mutation-based algorithm. Furthermore, the algorithm turns out to be surprisingly robust since the large neighbourhood explored by the mutation operator does not disrupt the search.

A graph is called universal for a given graph class (or, equivalently, -universal) if it contains a copy of every graph in as a subgraph. The construction of sparse universal graphs for various classes has received a considerable amount of attention. There is particular interest in tight -universal graphs, that is, graphs whose number of vertices is equal to the largest number of vertices in a graph from . Arguably, the most studied case is that when is some class of trees. In this work, we are interested in (n,Δ), the class of all n-vertex trees with maximum degree at most Δ. We show that every n-vertex graph satisfying certain natural expansion properties is (n,Δ)-universal. Our methods also apply to the case when Δ is some function of n. Since random graphs are known to be good expanders, our result implies, in particular, that there exists a positive constant c such that the random graph G(n,cn−1/3log2n) is asymptotically almost surely (a.a.s.) universal for (n,O(1)). Moreover, a corresponding result holds for the random regular graph of degree cn2/3log2n. Another interesting consequence is the existence of locally sparse n-vertex (n,Δ)-universal graphs. For example, we show that one can (randomly) construct n-vertex (n,O(1))-universal graphs with clique number at most five. This complements the construction of Bhatt, Chung, Leighton and Rosenberg (1989), whose (n,Δ)-universal graphs with merely O(n) edges contain large cliques of size Ω(Δ). Finally, we show that random graphs are robustly (n,Δ)-universal in the context of the Maker–Breaker tree-universality game.

A standard bridge between automata theory and logic is provided by the notion ofcharacteristic formula. This paper investigates this problem for the class ofevent-recording automata (ERA), a subclass of timed automata in which clocks areassociated with actions and that enjoys very good closure properties. We first study theproblem of expressing characteristic formulae for ERA in Event-Recording Logic (ERL ), alogic introduced by Sorea to express event-based timed specifications. We prove that theconstruction proposed by Sorea for ERA without invariants is incorrect. More generally, weprove that timed bisimilarity cannot in general be expressed in ERL for the class of ERA ,and study under which conditions on ERA it can be. Then, we introduce the logicWTμ , a new logic for event-based timed specificationscloser to the timed logic ℒν that was introduced byLaroussinie, Larsen and Weise. We prove that it is strictly more expressive than ERL , andthat its model-checking problem against ERA is EXPTIME -complete. Finally, we providecharacteristic formulae constructions in WTμ forcharacterizing the general class of ERA up to timed (bi)similarity and study thecomplexity issues.