  We show that a coupling of non-colliding simple random walkers on the
complete graph on $n$ vertices can include at most $n - \log n$ walkers. This
improves the only previously known upper bound of $n-2$ due to Angel, Holroyd,
Martin, Wilson, and Winkler ({\it Electron.~Commun.~Probab.~18}, 2013). The
proof considers couplings of i.i.d.~sequences of Bernoulli random variables
satisfying a similar avoidance property, for which there is separate interest.
Our bound in this setting should be closer to optimal.
