  \newcommand{\rhomi}[1]{\widetilde{H}_{#1}} \newcommand{\rbeti}[1]{\beta_{#1}}
\newcommand{\kk}{\mathbf k} \newcommand{\dimk}{\dim_{\kk}}
  We show that algebraically shifting a pair of simplicial complexes weakly
increases their relative homology Betti numbers in every dimension.
  More precisely, let $\Delta(K)$ denote the algebraically shifted complex of
simplicial complex $K$, and let $\rbeti{j}(K,L)=\dimk \rhomi{j}(K,L;\kk)$ be
the dimension of the $j$th reduced relative homology group over a field $\kk$
of a pair of simplicial complexes $L \subseteq K$. Then $\rbeti{j}(K,L) \leq
\rbeti{j}(\Delta(K),\Delta(L))$ for all $j$.
  The theorem is motivated by somewhat similar results about Gr\"obner bases
and generic initial ideals. Parts of the proof use Gr\"obner basis techniques.
