  A technical ingredient in Faltings' original approach to p-adic comparison
theorems involves the construction of $K(\pi, 1)$-neighborhoods for a smooth
scheme X over a mixed characteristic dvr with a perfect residue field: every
point of X has an open neighborhood whose general fiber is a $K(\pi, 1)$ scheme
(a notion analogous to having a contractible universal cover). We show how to
extend this result to the logarithmically smooth case, which might help to
simplify some proofs in p-adic Hodge theory. The main ingredient of the proof
is a variant of a trick of Nagata used in his proof of the Noether
Normalization Lemma.
