Subalgebras of a polynomial ring with minimal Hilbert function

In a recent paper by Boij and Conca the upper and lower bounds for the Hilbert function of subalgebras of a polynomial ring are discussed. In this talk we will study subalgebras generated in degree two with minimal Hilbert function. These subalgebras are generated by strongly stable sets of monomials. To minimize the Hilbert function we want to firstly minimize the numbers of variables, and secondly the multiplicity of the algebra. This boils down to a purely combinatorial problem, as the multiplicity can be computed by counting the number of maximal north-east lattice paths in an diagram representing the strongly stable set.