Abstract: For fixed graphs F 1,…,F r , we prove an upper bound on the threshold function for the property that G(n, p) → (F 1,…,F r ) This establishes the 1-statement of a conjecture of Kohayakawa and Kreuter

Abstract: The Brownian separable permuton is a random probability measure on the unit square, which was introduced by Bassino, Bouvel, Feray, Gerin, Pierrot (2016) as the scaling limit of the diagram of the uniform separable permutation as size grows to infinity. We show that, almost surely, the permuton is the pushforward of the Lebesgue measure on the graph of a random measure-preserving function associated to a Brownian excursion whose strict local minima are decorated with i.i.d. signs. As a consequence, its support is almost surely totally disconnected, has Hausdorff dimension one, and enjoys self-similarity properties inherited from those of the Brownian excursion. The density function of the averaged permuton is computed and a connection with the shuffling of the Brownian continuum random tree is explored.

Abstract: The theta graph consists of two vertices joined by t vertex-disjoint paths, each of length . For fixed odd and large t, we show that the largest graph not containing has at most edges and that this is tight apart from the value of .

Abstract: The maximum size of an r-uniform hypergraph without a Berge cycle of length at least k has been determined for all k ≥ r + 3 by Furedi, Kostochka and Luo and for k < r (and k = r, asymptotically) by Kostochka and Luo. In this paper we settle the remaining cases: k = r + 1 and k = r + 2, proving a conjecture of Furedi, Kostochka and Luo.

Abstract: We prove an essentially sharp lower bound on the k-round distributional complexity of the k-step pointer chasing problem under the uniform distribution, when Bob speaks first. This is an improvement over Nisan and Wigderson’s lower bound, and essentially matches the randomized lower bound proved by Klauck. The proof is information-theoretic, and a key part of it is using asymmetric triangular discrimination instead of total variation distance; this idea may be useful elsewhere.

Abstract: Given graphs H1, H2, a graph G is (H1, H2) -Ramsey if, for every colouring of the edges of G with red and blue, there is a red copy of H1 or a blue copy of H2. In this paper we investigate Ramsey questions in the setting of randomly perturbed graphs. This is a random graph model introduced by Bohman, Frieze and Martin [8] in which one starts with a dense graph and then adds a given number of random edges to it. The study of Ramsey properties of randomly perturbed graphs was initiated by Krivelevich, Sudakov and Tetali [30] in 2006; they determined how many random edges must be added to a dense graph to ensure the resulting graph is with high probability (K3, Kt) -Ramsey (for t ≽ 3). They also raised the question of generalizing this result to pairs of graphs other than (K3, Kt). We make significant progress on this question, giving a precise solution in the case when H1 = Ks and H2 = Kt where s, t ≽ 5. Although we again show that one requires polynomially fewer edges than in the purely random graph, our result shows that the problem in this case is quite different to the (K3, Kt) -Ramsey question. Moreover, we give bounds for the corresponding (K4, Kt) -Ramsey question; together with a construction of Powierski [37] this resolves the (K4, K4) -Ramsey problem. We also give a precise solution to the analogous question in the case when both H1 = Cs and H2 = Ct are cycles. Additionally we consider the corresponding multicolour problem. Our final result gives another generalization of the Krivelevich, Sudakov and Tetali [30] result. Specifically, we determine how many random edges must be added to a dense graph to ensure the resulting graph is with high probability (Cs, Kt) -Ramsey (for odd s ≽ 5 and t ≽ 4). To prove our results we combine a mixture of approaches, employing the container method, the regularity method as well as dependent random choice, and apply robust extensions of recent asymmetric random Ramsey results.

Abstract: The inducibility of a graph H measures the maximum number of induced copies of H a large graph G can have. Generalizing this notion, we study how many induced subgraphs of fixed order k and size l a large graph G on n vertices can have. Clearly, this number is for l = 1.Our proof methods involve probabilistic tools, such as anti-concentration results relying on fourth moment estimates and Brun’s sieve, as well as graph-theoretic and combinatorial arguments such as Zykov’s symmetrization, Sperner’s theorem and various counting techniques.

Abstract: We give an efficient algorithm that, given a graph G and a partition V1,…,Vm of its vertex set, finds either an independent transversal (an independent set {v1,…,vm} in G such that has a small dominating set. A non-algorithmic proof of this result has been known for a number of years and has been used to solve many other problems. Thus we are able to give algorithmic versions of many of these applications, a few of which we describe explicitly here.

Abstract: Pitting corrosion is a very serious problem for mild steel when it comes in contact with the dilute sulfuric acid medium. Specialized corrosion inhibitors are essentially required to minimize pitting and uniform types of corrosion in mild steel. Most of the corrosion inhibitors discovered so far protects the mild steel from uniform type of corrosion. But pitting corrosion is more fatal than a uniform type of corrosion because it immediately makes mild steel unfit for use as leakage starts from the pit. The objective was to protect the mild steel alloys from pitting corrosion when comes in contact with dilute sulfuric acid by the use of organic corrosion inhibitor. Cetyl Trimethyl Ammonium Bromide (CTAB) is tested as a corrosion inhibitor for mild steel in 0.1 N H2SO4 as corroding medium at 25.0, 30.0 and 35.0°C by weight loss, electrochemical polarization, and Impedance spectroscopy methods. Surface study of corroded and un-corroded specimens of mild steel was carried out by Metallurgical Research Microscopy (MRM) and Scanning Electron Microscopy (SEM) techniques. Surface study confirms that the adsorption of CTAB takes place through nitrogen atom resulting in the formation of uniform, nonporous, passive film confirmed by decrease in Warburg Impedance (Zw), decrease in Faradaic current, increase in Capacitive current, an increase in charge transfer resistance, Rct (41 to 401 Ω cm2) and significant increase in capacitive loop in Nyquist plot with increase in concentration of CTAB which results in significant decrease in corrosion rate of mild steel in 0.1N H2SO4 medium (percentage corrosion inhibition efficiency: 95.0%) especially eradicating pitting type of corrosion. CTAB was proved to be a very good anti-pit agent for mild steel in 0.1N sulfuric acid medium. Pitting and uniform type of corrosion was significantly reduced by the use of CTAB as corrosion inhibitor for mild steel in the dilute sulfuric acid medium at 25.0, 30.0 and 35.0°C.

Abstract: Random constraint satisfaction problems play an important role in computer science and combinatorics. For example, they provide challenging benchmark examples for algorithms, and they have been harnessed in probabilistic constructions of combinatorial structures with peculiar features. In an important contribution (Krzakala et al. 2007, Proc. Nat. Acad. Sci.), physicists made several predictions on the precise location and nature of phase transitions in random constraint satisfaction problems. Specifically, they predicted that their satisfiability thresholds are quite generally preceded by several other thresholds that have a substantial impact both combinatorially and computationally. These include the condensation phase transition, where long-range correlations between variables emerge, and the reconstruction threshold. In this paper we prove these physics predictions for a broad class of random constraint satisfaction problems. Additionally, we obtain contiguity results that have implications for Bayesian inference tasks, a subject that has received a great deal of interest recently (e.g. Banks et al. 2016, Proc. 29th COLT).

Abstract: Akbari and Alipour [1] conjectured that any Latin array of order n with at least n2/2 symbols contains a transversal. For large n, we confirm this conjecture, and moreover, we show that n399/200 symbols suffice.

Abstract: A classical result of Erdős and, independently, of Bondy and Simonovits [3] says that the maximum number of edges in an n-vertex graph not containing C2k, the cycle of length 2k, is O(n1+1/k). Simonovits established a corresponding supersaturation result for C2k’s, showing that there exist positive constants C,c depending only on k such that every n-vertex graph G with e(G)⩾ Cn1+1/k contains at least c(e(G)/v(G))2k copies of C2k, this number of copies tightly achieved by the random graph (up to a multiplicative constant).In this paper we extend Simonovits' result to a supersaturation result of r-uniform linear cycles of even length in r-uniform linear hypergraphs. Our proof is self-contained and includes the r = 2 case. As an auxiliary tool, we develop a reduction lemma from general host graphs to almost-regular host graphs that can be used for other supersaturation problems, and may therefore be of independent interest.

Abstract: A celebrated theorem of Pippenger states that any almost regular hypergraph with small codegrees has an almost perfect matching. We show that one can find such an almost perfect matching which is ‘pseudorandom’, meaning that, for instance, the matching contains as many edges from a given set of edges as predicted by a heuristic argument.

Abstract: The Hamming graph H(d, n) is the Cartesian product of d complete graphs on n vertices. Let be the degree and be the number of vertices of H(d, n). Let pc(d) be the critical point for bond percolation on H(d, n). We show that, for d ∈ ℕ fixed and n → ∞, pc(d) = 1/m + 2d2 - 1/2(d -1)2 1/m2 + O (m-3)+O (m-1V-1/3), which extends the asymptotics found in [10] by one order. The term O(m-1V-1/3) is the width of the critical window. For d=4,5,6 we have & m-3 = O(m-1V-1/3), and so the above formula represents the full asymptotic expansion of pc(d). In [16] we show that this formula is a crucial ingredient in the study of critical bond percolation on H(d, n) for. The proof uses a lace expansion for the upper bound and a novel comparison with a branching random walk for the lower bound. The proof of the lower bound also yields a refined asymptotics for the susceptibility of a subcritical Erdos-Renyi random graph.

Abstract: Given graphs G and H, a family of vertex-disjoint copies of H in G is called an H-tiling. Conlon, Gowers, Samotij and Schacht showed that for a given graph H and a constant γ>0, there exists C>0 such that if , then asymptotically almost surely every spanning subgraph G of the random graph 𝒢(n, p) with minimum degree at least contains an H-tiling that covers all but at most γn vertices. Here, χcr(H) denotes the critical chromatic number, a parameter introduced by Komlos, and m2(H) is the 2-density of H. We show that this theorem can be bootstrapped to obtain an H-tiling covering all but at most vertices, which is strictly smaller when . In the case where H = K3, this answers the question of Balogh, Lee and Samotij. Furthermore, for an arbitrary graph H we give an upper bound on p for which some leftover is unavoidable and a bound on the size of a largest H -tiling for p below this value.

Abstract: Monotonic surfaces spanning finite regions of ℤd arise in many contexts, including DNA-based self-assembly, card-shuffling and lozenge tilings. One method that has been used to uniformly generate these surfaces is a Markov chain that iteratively adds or removes a single cube below the surface during a step. We consider a biased version of the chain, where we are more likely to add a cube than to remove it, thereby favouring surfaces that are ‘higher’ or have more cubes below it. We prove that the chain is rapidly mixing for any uniform bias in ℤ2 and for bias λ > d in ℤd when d > 2. In ℤ2 we match the optimal mixing time achieved by Benjamini, Berger, Hoffman and Mossel in the context of biased card shuffling [2], but using much simpler arguments. The proofs use a geometric distance function and a variant of path coupling in order to handle distances that can be exponentially large. We also provide the first results in the case of fluctuating bias, where the bias can vary depending on the location of the tile. We show that the chain continues to be rapidly mixing if the biases are close to uniform, but that the chain can converge exponentially slowly in the general setting.

Abstract: We prove Bogolyubov–Ruzsa-type results for finite subsets of groups with small tripling, |A 3| ≤ O(|A|), or small alternation, |AA −1A| ≤ O(|A|). As applications, we obtain a qualitative analogue of Bogolyubov’s lemma for dense sets in arbitrary finite groups, as well as a quantitative arithmetic regularity lemma for sets of bounded VC-dimension in finite groups of bounded exponent. The latter result generalizes the abelian case, due to Alon, Fox and Zhao, and gives a quantitative version of previous work of the author, Pillay and Terry.

Abstract: работе проведен процесс коксования тяжелого нефтяного остатка – гудрона Павлодарского нефтехимического завода с целью снижения содержания металлов и серы и повышения выхода кокса. Предварительно проводится деметаллизация и обессеривание гудрона адсорбентами на основе цеолита, модифицированного ксерогелью оксида ванадия (V). В результате процесса коксования выход кокса составил 22-36 %, по основным физико-химическим показателям образцы полученного кокса удовлетворяют требованиям марки КЗА.

Abstract: For an integer q ⩾ 2, a graph G is called q-Ramsey for a graph H if every q-colouring of the edges of G contains a monochromatic copy of H. If G is q-Ramsey for H yet no proper subgraph of G has this property, then G is called q-Ramsey-minimal for H. Generalizing a statement by Burr, Nesetřil and Rodl from 1977, we prove that, for q ⩾ 3, if G is a graph that is not q-Ramsey for some graph H, then G is contained as an induced subgraph in an infinite number of q-Ramsey-minimal graphs for H as long as H is 3-connected or isomorphic to the triangle. For such H, the following are some consequences. For 2 ⩽ r < q, every r-Ramsey-minimal graph for H is contained as an induced subgraph in an infinite number of q-Ramsey-minimal graphs for H. For every q ⩾ 3, there are q-Ramsey-minimal graphs for H of arbitrarily large maximum degree, genus and chromatic number. The collection forms an antichain with respect to the subset relation, where denotes the set of all graphs that are q-Ramsey-minimal for H. We also address the question of which pairs of graphs satisfy , in which case H 1 and H 2 are called q-equivalent. We show that two graphs H 1 and H 2 are q-equivalent for even q if they are 2-equivalent, and that in general q-equivalence for some q ⩾ 3 does not necessarily imply 2-equivalence. Finally we indicate that for connected graphs this implication may hold: results by Nesetřil and Rodl and by Fox, Grinshpun, Liebenau, Person and Szabo imply that the complete graph is not 2-equivalent to any other connected graph. We prove that this is the case for an arbitrary number of colours.

Abstract: We show that a dense subset of a sufficiently large group multiplication table contains either a large part of the addition table of the integers modulo some k, or the entire multiplication table of a certain large abelian group, as a subgrid. As a consequence, we show that triples systems coming from a finite group contain configurations with t triples spanning vertices, which is the best possible up to the implied constant. We confirm that for all t we can find a collection of t triples spanning at most t + 3 vertices, resolving the Brown–Erdős–Sos conjecture in this context. The proof applies well-known arithmetic results including the multidimensional versions of Szemeredi’s theorem and the density Hales–Jewett theorem. This result was discovered simultaneously and independently by Nenadov, Sudakov and Tyomkyn [5], and a weaker result avoiding the arithmetic machinery was obtained independently by Wong [11].

Abstract: Being by-products of non-ferrous metallurgy, slags contain fayalite (Fe2SiO4) as the major component. Since hydrometallurgical methods are considered as the most promising for processing such material to obtain valuable metals, increasing the leachability of fayalite in sulfuric acid as a widely used leaching agent is an important task. The present work was devoted to increasing the reactivity of fayalite by using mechanical activation. Fayalite, synthesized with the use of powders of metallic Fe, Fe2O3, and SiO2, was subjected to mechanical activation in the planetary ball mill at 400 rpm with a ball/powder ratio of 5 for 45 minutes. Then, activated and non-activated fayalite samples were subjected to sulfuric acid leaching. Before leaching, solid samples were characterized by XRD and Dynamic Light Scattering (DLS). Quantitative analysis of Fe and Si in the leachate was determined by Inductively Coupled Plasma-Atomic Emission Spectroscopy. Mechanical activation led to partial amorphization of the initial fayalite sample. It was found that the leaching rate constants of the treated samples in sulfuric acid solution (50-80 g×L-1) at 298, 338, and 368 K increased and the activation energy of the leaching process decreased, i.e. mechanical activation enhances the reactivity of fayalite in H2SO4 solution. Mechanical activation can be applied to improve fayalite leachability in sulfuric acid solution. The results obtained can be used in the development of methods for leaching slag of non-ferrous metallurgy, in particular, copper smelter slags, the major component of which is fayalite.

Abstract: We study structural properties of graphs with bounded clique number and high minimum degree. In particular, we show that there exists a function L = L(r,ɛ) such that every Kr-free graph G on n vertices with minimum degree at least ((2r–5)/(2r–3)+ɛ)n is homomorphic to a Kr-free graph on at most L vertices. It is known that the required minimum degree condition is approximately best possible for this result.For r = 3 this result was obtained by Łuczak (2006) and, more recently, Goddard and Lyle (2011) deduced the general case from Łuczak’s result. Łuczak’s proof was based on an application of Szemeredi’s regularity lemma and, as a consequence, it only gave rise to a tower-type bound on L(3, ɛ). The proof presented here replaces the application of the regularity lemma by a probabilistic argument, which yields a bound for L(r, ɛ) that is doubly exponential in poly(ɛ).

Abstract: An equitable colouring of a graph $G$ is a colouring of the vertices of $G$ so that no two adjacent vertices are coloured the same and, additionally, the colour class sizes differ by at most $1$. The equitable chromatic number $\chi_=(G)$ is the minimum number of colours required for this. We study the equitable chromatic number of the dense random graph $G(n,m)$, where $m = \left\lfloor p {n \choose 2} \right \rfloor $ and $0