The proposed research areas are extremal graph theory and bootstrap percolation. They are not far from each other, as many probabilistic tools are used in the first one, and many combinatorial ideas are needed in the second one. The need of computer science and demands from applications where discrete models play more and more important roles, increase the importance of extremal graph theory and suggests an algorithmic point of view. For about forty years now, percolation theory has been an active area of research at the interface of probability theory, combinatorics and physics. Interest in various aspects of standard percolation remains high, including estimates of critical probabilities. Lately more and more variants of the standard percolation models have been studied, in particular, the family of processes known as bootstrap percolation. Recent applications arise from different aspects, for example from spatio-temporal dynamical systems. Computer experiments performed by physicists have suggested interesting non-trivial large-scale behavior, and many deep mathematical results have been proved about a number of models. The proposer is aiming to study the percolation process at the critical probability.
The work of the proposer is an extension of Turan's Theorem into several directions. One direction is to describe graph families which do not contain certain induced subgraphs. The other is to study Turan type of questions on hypergraphs, in particular on triple systems, and to develop general tools like regularity and stability theorems. Bootstrap percolation, a member of the family of random cellular automata, is a process on graphs, where each site is open or closed with a certain probability, and these states are changing with time. Studying bootstrap percolation, the main aim of the proposer is to describe the phase transition, estimate the critical probability, and the size of the window around the critical probability. The plan is to prove that the transitions are sharp, and to investigate different models, whose understanding would be helpful in the applications.
