For a simplicial complex 2 and coefficient domain F let F2 be the F-module with basis 2. We investigate the inclusion map given by : { [ _ 1 +_ 2 +_ 3 + } } } +_ k which assigns to every face { the sum of the co-dimension 1 faces contained in {. When the coefficient domain is a field of characteristic p>0 this map gives rise to homological sequences. We investigate this modular homology for shellable complexes. Academic Press

Among shellable complexes a certain class is shown to have maximal modular homology, and these are the so-called saturated complexes. We show that certain conditions on the links of the complex imply saturation. We prove that Coxeter complexes and buildings are saturated. © 2002 Elsevier Science (USA)

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Let $G$ be a permutation group on the set $\Omega$ and let ${\cal S}$ be a collection of subsets of $\Omega,$ all of size $\geq m$ for some integer $m$. For $s\leq m$ let $n_{s}(G,\,{\cal S})$ be the number of $G$-orbits on the subsets of $\Omega$ which have a representative $y\subseteq x$ with $|y|=s$ and $y\subseteq x$ for some $x\in {\cal S}$. We prove that if $s < t$ with $s+t\leq m$ then $n_{s}(G,\,{\cal S})\leq n_{t}(G,\,{\cal S})$. A special case of this theorem is the Livingstone-Wagner Theorem when ${\cal S}=\{\Omega\}$. We show how the result can be applied to estimate orbit numbers for simplicial complexes, sequences, graphs and amalgamation classes. It is also shown how this theorem can be extended to orbit theorems on more general partially ordered sets.

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