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## Random hypergraphs and property B

In 1964 Erdos proved that $(1+\oh{1})) \frac{\eul \ln(2)}{4} k^2 2^{k}$ edges are sufficient to build a $k$-graph which is not two colorable. To this day, it is not known whether there exist such $k$-graphs with smaller number of edges.

Erd\H{o}s' bound is consequence of the fact that a hypergraph with $k^2/2$ vertices and $M(k)=(1+\oh{1}) \frac{\eul \ln(2)}{4} k^2 2^{k}$ randomly chosen edges of size $k$ is not two colorable with high probability.

Our first main result implies that for any $\varepsilon > 0$, any $k$-graph with $(1-\varepsilon) M(k)$ randomly and uniformly chosen edges is a.a.s. two colorable.

The presented proof is an adaptation of the second moment method analogous to the developments of Achlioptas and Moore from 2002 who considered the problem with fixed size of edges and number of vertices tending to infinity.

In the second part of the paper we consider the problem of algorithmic coloring of random $k$-graphs.

We show that quite simple, and somewhat greedy procedure, a.a.s. finds a proper coloring for random $k$-graphs on $k^2/2$ vertices, with at most $\Oh{k\ln k\cdot 2^k}$ edges. That value coincides with the analogue of algorithmic barrier defined by Achlioptas and Coja-Oghlan in 2008, for the case of fixed $k$.

The paper deals with estimating the r-colorability threshold for a random k-uniform hypergraph in the binomial model H(n,k,p). We consider the sparse case, when the expected number of edges is a linear function of n and prove a new lower bound for the sharp threshold of the property that H(n,k,p) is r-colorable.

Different ways of the visual model descriptions formalization are considered. The hypergraph with poles is offered as a new formal model for graphic languages creation. This model provides possibility of definition and implementation of new visual languages and it gives a basis for realization of operations over the models constructed with these languages. The suggested model is an extension of the graph with poles concept for considering specifics of graphics editors implementation for the DSM platforms. The language toolkits include means of languages definition and tools for models creation, but components for model transformations are the most important in modern DSM platform. New formal model provides flexible tools for decomposition and detailing of models and language transformations feasibility.

The paper deals with weak chromatic numbers of random hypergraphs. Recall that a vertex coloring is said to be j-proper for a hypergraph if every j+1 vertices of any edge do not share a common color. The j-chromatic number of a hypergraph is the minimum number of colors required for a -proper coloring. We study the j-chromatic number of a random hypergraph in the binomial model H(n,k,p) in the case j=k-2 and investigate, for fixed k, the threshold for the property that j-chromatic number of does not exceed r. This threshold corresponds to the sparse case and the main result gives the tight bounds for it.

We study the asymptotic behavior of probabilities of first-order properties for random uniform hypergraphs. In 1990, J. Spencer introduced the notion of a spectrum for graph properties and proved the existence of a first-order property with an infinite spectrum. In this paper we give a definition of a spectrum for properties of uniform hypergraphs and establish an almost tight bound for the minimum quantifier depth of a first-order formula with infinite spectrum.

A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.