The investigator works in three areas of Combinatorics: Probabilistic Methods, Extremal Combinatorics, and Ramsey Theory. Using the methodology of the probabilistic method, as started by Paul Erdos, the existence of a combinatorial object is shown by exhibiting a suitable probabilistic space in which the desired object has a positive probability. One of the goals is to find efficient methods for studying probabilistic spaces arising from combinatorial problems. The first part of this project concentrates on two lines of research: on a detailed description of the threshold phenomena mainly in the framework of the first order logics and on research in the area of 0-1 laws for logics with richer signatures. Due to the methods used this area lies on the border between probabilistic methods and finite model theory. The investigator's focus in extremal combinatorics and ramsey theory is on studying set systems with forbidden configurations. This is, very often, closely connected with the research in probabilistic methods. An example of problems in this area, which connect extremal combinatorics, ramsey theory, as well as information theory, are problems on delta-systems and on Shannon capacities of graphs.
The investigator works in the area of Combinatorics. One part of this project concentrates on studying random combinatorial structures. This is motivated by problems coming from the area of random algorithms as well as from the area studying random models in statistical mechanics. As an example, the investigator studies emergence of certain properties in random structures, so called threshold phenomena. The research involves developing efficient methods for studying probabilistic spaces arising in these problems as well as studying logics which describe the properties of interest. The investigator's focus in extremal combinatorics and ramsey theory is on studying set systems with forbidden configurations. Equivalently, the problems studied by the investigator deal with appearance of certain patterns in set systems. An example of problems in this area are problems on delta-systems and on Shannon capacities of graphs. These problems are connected to problems in information theory and transmission of information.
