Applications of the graph container method

In this talk we plan to have a gentle introduction of the basic graph container method and the advanced Saphozenko graph container algorithm. We shall discuss counting and probabilistic questions on the number of antichains of the Boolean lattice, and on the maximum size of an antichain of a random subset of the Boolean lattice. Similarly, we are interested in the number and typical structure of independent sets in the (random subset) of the hypercube. The methods easily extend to d-regular n-vertex graphs, where 1<< d =O(polylog n).