**MSC:**- 05B99 None of the above but in this section
- 05C35 Extremal problems, See also {90C35}

maximum number of covering pairs achieved by an $n$-element poset

in ${\cal F}$. We determine this number of the class of $N$-free

lattices

and give crude lower and upper bounds for the classes of ditributive

and covering lattices. The structural characterization of covering

lattices leads us to a class of lattices whose elements have a boundet

number of lower and upper covers. This class statisfies a conjecture

of Bollobas and RRival, who originally, ask for $ex({\cal L}, n)$

where

${\cal L}$ denotes the class of all lattices.