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A relative simplicial complex is a collection of sets of the form $\Delta\setminus \Gamma$, where $\Gamma \subset \Delta$ are simplicial complexes.Relative complexes played key roles in recent advances in algebraic, geometric,and topological combinatorics but, in contrast to simplicial complexes, littleis known about their general combinatorial structure. In this paper, we addressa basic question in this direction and give a characterization of $f$-vectorsof relative (multi)complexes on a ground set of fixed size. On the algebraicside, this yields a characterization of Hilbert functions of quotients ofhomogeneous ideals over polynomial rings with a fixed number of indeterminates.  Moreover, we characterize $h$-vectors of fully Cohen--Macaulay relativecomplexes as well as $h$-vectors of Cohen--Macaulay relative complexes withminimal faces of given dimensions. The latter resolves a question of Bj\"orner.

On f- and h- vectors of relative simplicial complexes