  Answering a question by Angel, Holroyd, Martin, Wilson and Winkler, we show
that the maximal number of non-colliding coupled simple random walks on the
complete graph $K_N$, which take turns, moving one at a time, is monotone in
$N$. We use this fact to couple $\lceil \frac N4 \rceil$ such walks on $K_N$,
improving the previous $\Omega(N/\log N)$ lower bound of Angel et al. We also
introduce a new generalization of simple avoidance coupling which we call
partially ordered simple avoidance coupling and provide a monotonicity result
for this extension as well.

Probability