Towards a Katona Type Proof for the 2-intersecting Erdos-Ko-Rado Theorem

Towards a Katona type proof for the 2-intersecting Erdös-Ko-Rado theorem

Erdös-Ko-Rado from intersecting shadows

A set system is called t-intersecting if every two members meet each other in at least t elements. Katona determined the minimum ratio of the shadow and the size of such families and showed that the Erdős–Ko–Rado theorem immediately follows from this result. The aim of this note is to reproduce the proof to obtain a slight improvement in the Kneser graph. We also give a brief overview of corres...

Note Erdős–Ko–Rado from intersecting shadows

A set system is called t-intersecting if every two members meet each other in at least t elements. Katona determined the minimum ratio of the shadow and the size of such families and showed that the Erdős– Ko–Rado theorem immediately follows from this result. The aim of this note is to reproduce the proof to obtain a slight improvement in the Kneser graph. We also give a brief overview of corre...

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...

Erdos-Ko-Rado in Random Hypergraphs

Let 3 ≤ k < n/2. We prove the analogue of the Erdős-Ko-Rado theorem for the random k-uniform hypergraph Gk(n, p) when k < (n/2)1/3; that is, we show that with probability tending to 1 as n → ∞, the maximum size of an intersecting subfamily of Gk(n, p) is the size of a maximum trivial family. The analogue of the Erdős-Ko-Rado theorem does not hold for all p when k À n1/3. We give quite precise r...