  We show that every countable Borel equivalence relation structurable by
$n$-dimensional contractible simplicial complexes embeds into one which is
structurable by such complexes with the further property that each vertex
belongs to at most $M_n := 2^{n-1}(n^2+3n+2)-2$ edges; this generalizes a
result of Jackson-Kechris-Louveau in the case $n = 1$. The proof is based on
that of a classical result of Whitehead on countable CW-complexes.
