Mitschke’s theorem is sharp

Mitschke showed that a variety with an m-ary near-unanimity term has Jonsson terms t(0), ..., t(2m-4) witnessing congruence distributivity. We show that Mitschke's result is sharp. We also evaluate the best possible number of Day terms witnessing congruence modularity. More generally, we characterize exactly the best bounds for many congruence identities satisfied by varieties with an m-ary near-unanimity term. Finally we present some simple observations about terms with just one "dissenter", a generalization of a minority term.