We say that α ∈ [0, 1) is a jump for an integer r ≥ 2 if there exists c(α) > 0 such that for all ϵ > 0 and all t ≥ 1, any r-graph with n ≥ n0(α, ϵ, t) vertices and density at least α + ϵ contains a subgraph on t vertices of density at least α + c.

The Erdős–Stone–Simonovits theorem [4, 5] implies that for r = 2, every α ∈ [0, 1) is a jump. Erdős [3] showed that for all r ≥ 3, every α ∈ [0, r!/rr) is a jump. Moreover he made his famous ‘jumping constant conjecture’, that for all r ≥ 3, every α ∈ [0, 1) is a jump. Frankl and Rödl [7] disproved this conjecture by giving a sequence of values of non-jumps for all r ≥ 3.

We use Razborov's flag algebra method [9] to show that jumps exist for r = 3 in the interval [2/9, 1). These are the first examples of jumps for any r ≥ 3 in the interval [r!/rr, 1). To be precise, we show that for r = 3 every α ∈ [0.2299, 0.2316) is a jump.

We also give an improved upper bound for the Turán density of K4− = {123, 124, 134}: π(K4−) ≤ 0.2871. This in turn implies that for r = 3 every α ∈ [0.2871, 8/27) is a jump.

In this paper we consider the influences of variables on Boolean functions in general product spaces. Unlike the case of functions on the discrete cube, where there is a clear definition of influence, in the general case several definitions have been presented in different papers. We propose a family of definitions for the influence that contains all the known definitions, as well as other natural definitions, as special cases. We show that the proofs of the BKKKL theorem and of other results can be adapted to our new definition. The adaptation leads to generalizations of these theorems, which are tight in terms of the definition of influence used in the assertion.

The asymptotics of 2-colour Ramsey numbers of loose and tight cycles in 3-uniform hypergraphs were recently determined [16, 17]. We address the same problem for Berge cycles and for 3 colours. Our main result is that the 3-colour Ramsey number of a 3-uniform Berge cycle of length n is asymptotic to . The result is proved with the Regularity Lemma via the existence of a monochromatic connected matching covering asymptotically 4n/5 vertices in the multicoloured 2-shadow graph induced by the colouring of Kn(3).

Let I be an independent set drawn from the discrete d-dimensional hypercube Qd = {0, 1}d according to the hard-core distribution with parameter λ > 0 (that is, the distribution in which each independent set I is chosen with probability proportional to λ|I|). We show a sharp transition around λ = 1 in the appearance of I: for λ > 1, min{|I ∩ Ɛ|, |I ∩ |} = 0 asymptotically almost surely, where Ɛ and are the bipartition classes of Qd, whereas for λ < 1, min{|I ∩ Ɛ|, |I ∩ |} is asymptotically almost surely exponential in d. The transition occurs in an interval whose length is of order 1/d.

A key step in the proof is an estimation of Zλ(Qd), the sum over independent sets in Qd with each set I given weight λ|I| (a.k.a. the hard-core partition function). We obtain the asymptotics of Zλ(Qd) for , and nearly matching upper and lower bounds for , extending work of Korshunov and Sapozhenko. These bounds allow us to read off some very specific information about the structure of an independent set drawn according to the hard-core distribution.

We also derive a long-range influence result. For all fixed λ > 0, if I is chosen from the independent sets of Qd according to the hard-core distribution with parameter λ, conditioned on a particular v ∈ Ɛ being in I, then the probability that another vertex w is in I is o(1) for w ∈ but Ω(1) for w ∈ Ɛ.

In a previous paper, we have described the construction of anautomaton from a rational expression which has the property thatthe automaton built from an expression which is itself computedfrom a co-deterministic automaton by the state elimination methodis co-deterministic. It turned out that the definition on which the construction isbased was inappropriate, and thus the proof of the property wasflawed. We give here the correct definition of the broken derived termsof an expression which allow to define the automaton and thedetailed full proof of the property.

A word u defined over an alphabet $\mathcal A$ is c-balanced (c∈ $\mathbb N$ ) if for all pairs of factors v, w of u of the same lengthand for all letters a∈ $\mathcal A$ , the difference between the number of letters a in v and w is less or equal to c. In this paper we consider a ternary alphabet $\mathcal A$ = {L, S, M} and a class of substitutions $\varphi_p$ defined by $\varphi_p$ (L) = LpS, $\varphi_p$ (S) = M, $\varphi_p$ (M) = Lp–1S where p> 1.We prove that the fixed point of $\varphi_p$ , formally written as $\varphi_p^\infty$ (L), is 3-balanced and that its Abelian complexity is bounded above by the value 7, regardless of the value of p. We also show that both these bounds are optimal, i.e. they cannot be improved.

Analyzing genomic data for finding those gene variations which are responsible for hereditary diseases is one of the great challenges in modern bioinformatics. In many living beings (including the human), every gene is present in two copies, inherited from the two parents, the so-called haplotypes. In this paper, we propose a simple combinatorial model for classifying the set of haplotypes in a population according to their responsibility for a certain genetic disease. This model is based on the minimum-ones 2SAT problem with uniform clauses.The minimum-ones 2SAT problem asks for a satisfying assignment to a satisfiable formula in 2CNF which sets a minimum number of variables to true. This problem is well-known to be $\mathcal{NP}$ -hard, even in the case where all clauses are uniform, i.e., do not contain a positive and a negative literal. We analyze the approximability and present the first non-trivial exact algorithm for the uniform minimum-ones 2SAT problem with a running time of $\mathcal{O}$ (1.21061n ) on a 2SAT formula with n variables. We also show that the problem is fixed-parameter tractable by showing that our algorithm can be adapted to verify in $\mathcal{O}^*$ (2k ) time whether an assignment with at most k true variables exists.

A modified version of the classical µ -operator as well as thefirst value operator and the operator of inverting unaryfunctions, applied in combination with the composition offunctions and starting from the primitive recursive functions,generate all arithmetically representable functions. Moreover, thenesting levels of these operators are closely related to thestratification of the arithmetical hierarchy. The same is shownfor some further function operators known from computability and complexitytheory. The close relationships between nesting levels of operators andthe stratification of the hierarchy also hold for suitablerestrictions of the operators with respect to the polynomialhierarchy if one starts with the polynomial-time computablefunctions. It follows that questions around P vs. NP andNP vs. coNP can equivalently be expressed by closureproperties of function classes under these operators. The polytime version of the first value operator can be used toestablish hierarchies between certain consecutive levels withinthe polynomial hierarchy of functions, which are related togeneralizations of the Boolean hierarchies over the classes $\mbox{$\Sigma^p_{k}$}$ .

We introduce two graph polynomials and discuss their properties. One is a polynomial of two variables whose investigation is motivated by the performance analysis of the Bethe approximation of the Ising partition function. The other is a polynomial of one variable that is obtained by the specialization of the first one. It is shown that these polynomials satisfy deletion–contraction relations and are new examples of the V-function, which was introduced by Tutte (Proc. Cambridge Philos. Soc.43, 1947, p. 26). For these polynomials, we discuss the interpretations of special values and then obtain the bound on the number of sub-coregraphs, i.e., spanning subgraphs with no vertices of degree one. It is proved that the polynomial of one variable is equal to the monomer–dimer partition function with weights parametrized by that variable. The properties of the coefficients and the possible region of zeros are also discussed for this polynomial.