An order-convex set in a partially ordered set is a subset 
such that 
and 
imply 
. In other words, the order interval between any two comparable
members of 
is contained in 
.
Every antichain is order-convex, since no two distinct elements of an antichain are comparable. In a totally ordered set, order-convex sets are intervals, with any choice of included or excluded endpoints that exists in the order. In a Boolean lattice ordered by inclusion, order-convexity requires that every set between two comparable members also belongs to the family.
The Daykin-Frankl conjecture concerns the minimum possible partial order width of an order-convex family in a Boolean lattice.
