A new short proof for the Kruskal-Katona theorem

A new short proof for the Kruskal-Katona theorem

A Kruskal - Katona Type Theorem for the

We present an analog of the well-known Kruskal-Katona theorem for the poset of subspaces of PG (n; 2) ordered by inclusion. For given k; ` (k < `) and m the problem is to nd a family of size m in the set of`-subspaces of PG (n; 2), containing the minimal number of k-subspaces. We introduce two lexicographic type orders O 1 and O 2 on the set of`-subspaces, and prove that the rst m of them, take...

A Kruskal-Katona type theorem for graphs

A bound on consecutive clique numbers of graphs is established. This bound is evaluated and shown to often be much better than the bound of the Kruskal-Katona theorem. A bound on non-consecutive clique numbers is also proven.

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 Krus...

Approximations to the Kruskal-katona Theorem

Approximations to the Kruskal-Katona theorem are stated and proven. These approximations are weaker than the theorem, but much easier to work with numerically.