  We examine the m-shades of t-intersecting families of k-subsets of [n], and
conjecture on the optimal upper bound on their cardinalities. This conjecture
extends Frankl's General Conjecture that was proven true by
Ahlswede-Khachatrian. From this we deduce the precise asymptotic upper bounds
on the cardinalities of m-shades of t(m)-intersecting families of k(m)-subsets
of [2m], as m -> infinity. A generalization to cross-t-intersecting families is
also considered.
