  We generalize several classical theorems in extremal combinatorics by
replacing a global constraint with an inequality which holds for all objects in
a given class. In particular we obtain generalizations of Tur\'an's theorem,
the Erd\H{o}s-Gallai theorem, the LYM-inequality, the Erd\H{o}s-Ko-Rado theorem
and the Erd\H{o}s-Szekeres theorem on sequences.
