• f-vectors
• Kruskal-Katona Theorem
• Macaulay Theorem
• relative multicomplexes
• relative simplicial complexes

The focus of this work is studying f-vectors in a relative setting. The Kruskal-Katona theorem is a fundamental theorem in Combinatorics which characterizes f-vectors of simplicial complexes. A similar theorem for multicomplexes is the Macauley theorem, which also has a very natural formulation in terms of Hilbert functions of standard graded K-algebras. In my thesis I consider the question of characterizing f-vectors of relative simplicial complexes. A relative simplicial complex is a collection of sets given as the set-theoretic difference between a simplicial complex and a subcomplex. I obtain combinatorial and algebraic generalizations of the Kruskal-Katona and Macaulay characterizations under certain conditions on the number of vertices of simplicial complexes constituting the relative complex.