Shifting shadows: the Kruskal–Katona Theorem

As we have seen, antichains and intersecting families are fundamental to Extremal Set Theory. The two central theorems, Sperner’s Theorem and the Erdős–Ko–Rado Theorem, have inspired decades of research since their discovery, helping establish Extremal Set Theory as a vibrant and rapidly growing area of Discrete Mathematics. One must, then, pay a greater than usual amount of respect to the Kruskal–Katona Theorem, as it builds on both Sperner’s Theorem and the Erdős–Ko–Rado Theorem. Indeed, as you will prove for your homework assignment, the Kruskal–Katona Theorem provides a precise and refined characterisation of the number of sets of different sizes that an antichain can contain. On the other hand, the Erdős–Ko–Rado Theorem follows almost immediately as a consequence of the Kruskal–Katona Theorem. There is no doubt, then, that the Kruskal–Katona Theorem is truly a gem of Extremal Set Theory. In this note we introduce the theorem and give a short proof via the shifting technique, a powerful method that can be used to prove many results in this field.