Limit theorems in the theory of random hypergraphs

The report is devoted to one of the central directions of the probabilistic combinatorics — theory of random hypergraphs. The classical binomial model of a random hypergraph H(n,k,p), where each k-subset (edge) of the set of n vertices is included in a random hypergraph independently of the others with probability p, is the main object of the analysis. The report briefly highlights the history of the research and the new results obtained by the author. The report consists of two parts. In the first part of the report, we will discuss the limit behavior of the independence number of the random hypergraph H(n,k,p) and its parametric generalizations, j-independence number. The most incomplete case in this area is the case of a sparse random hypergraph, when the expectation of the number of edges is linear with respect to the number of vertices. Here the author established the laws of large numbers for the j-independence numbers in the form of the existence theorem. For a strongly sparse case, it is even possible to indicate the explicit form of the limit constant. The second big topic considers colorings of random hypergraphs. Here the limit distribution of the j-chromatic number of a random hypergraph in a sparse case has been studied. The first new result gives very accurate estimates of the threshold probability for the property of j-chromatic number equals two. The second new result in this direction establishes the limit distribution of the j-chromatic number of the random hypergraph H(n,k,p) in the case j=k-2. In particular, it is shown that the set of limit values is either one-point or consists of two adjacent values which can be found explicitly.