A simple proof of the Kruskal-Katona theorem

Continuous versions of some extremal hypergraph problems. II

“…Choose h=g-1 and define (p by go(E)= {(xl, ..., xo_l):(xl, ..., xo)CE }. Then for a graph G=(X,E) go(E) denotes the set of ("oriented") non-loop (g-1)-tuples being subsets of some edge in E. It is known (see [2], [3], simple proofs: [10], [1 I] [12]) that (Ngg) non-oriented g-edges contain at least ""'.. It is easy to see that the latter inequalities can be satisfied with Nl=nallo.-t-ol(n) and N,=nal/g+o2(n).…”

“…A good representative of this class is the following theorem of KRUS~AL [2]. (By some kind authors it is called Kruskal--Katona theorem, based on [3]; for simple proofs see [10], [11] and [12]. ): Given the number of vertices and edges (g-tuples with a fixed g) of the hypergraph, the theorem determines the minimum number of (g-1)-tuples contained by at least one edge.…”

Erdős–Ko–Rado with conditions on the minimum complementary degree

“…The original proof of the HiltonMilner theorem was quite long and complicated. There are now many shorter proofs, one of the first by Daykin [5] and a fairly simple one (of just the "well-known" part) by Frankl and Füredi [10].…”

“…If n = 11 and k = 5 then the (n − k − 1)-cascade forms for 2 and 3 are 2 = b (5) (5,4) and 3 = b (5) (5,4,3). By inequalities (4.2) We can remove any f (11, 5, 2) − 2 − 200 = 1 set containing 1 from F (11,5,2) to get a set system H which is DCM, but obviously not MDCM, since it is not maximal. There is also a set system of the same size and maximum degree as H which is DCM, not MDCM, but is maximal intersecting: it contains the sets {2,3,4,5,6}, {2,3,4,7,8}, and the 200 5-sets containing 1 which intersect them both.…”

$$f$$ -Vectors Implying Vertex Decomposability

“…Daykin [4,5] gave two simple proofs, and later Hilton [9] gave another one. For an algebraic proof, we refer the reader to [1].…”