Let be the lower-shadow size prescribed by the binomial representation in the Kruskal-Katona theorem. ThenInduction using Pascal's identity proves the inequality and drives the standard ground-set induction for the theorem.
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The Kruskal-Katona theorem is a result in combinatorial set theory, particularly related to the theory of hypergraphs and the study of families of sets. It provides a connection between the structure of a family of sets and the number of its intersections. The theorem defines conditions under which an antipodal family (a family of subsets) can be characterized in terms of its lower shadow, which is a fundamental concept in combinatorics.