An ordered r-graph is an r-uniform hypergraph whose vertex set linearly ordered. Given 2?k?r, H interval k-partite if there exist at least k disjoint intervals in the ordering such that every edge of has nonempty intersection with each and contained their union. Our main result implies ?>k?1, then for d>0 n-vertex dn? edges some m?n m-vertex subgraph ?(dm?) edges. This extension to r-graphs obs...

Let r ≥ 3 and (lnn)−1/(r−1) ≤ α ≤ r−3. We show that: Every r-uniform graph on n vertices with at least αnr/r! edges contains a complete r-partite graph with r − 1 parts of size ⌊ α (lnn)1/(r−1) ⌋ and one part of size ⌈ n1−α r−2 ⌉ . This result follows from a more general digraph version: Let U1, . . . , Ur be sets of size n, and M ⊂ U1 × · · · × Ur satisfy |M | ≥ αnr. If the integers s1, . . . ...

We show that the threshold cr,k for appearance of a k-core in a random r-partite r-uniform hypergraph Gr,n,m is the same as for a random r-uniform hypergraph with cn/r edges without the r-partite restriction, where r, k ≥ 2. In both cases, the average degree is c. This is an important problem in the analysis of the algorithm presented in [2]. The algorithm constructs a family of minimal perfect...

Let f(n, r, k) be the minimal number such that every hypergraph larger than f(n, r, k) contained in ([n] r ) contains a matching of size k, and let g(n, r, k) be the minimal number such that every hypergraph larger than g(n, r, k) contained in the r-partite r-graph [n]r contains a matching of size k. The Erdős-Ko-Rado theorem states that f(n, r, 2) = (n−1 r−1 ) (r ≤ n 2 ) and it is easy to show...

We strengthen the stability theorem of Erdős and Simonovits. Write Kr (s1, . . . , sr) for the complete r-partite graph with parts of size s1, . . . , sr and Tr (n) for the r-partite Turán graph of order n. Our main result is: For all r ≥ 2 and sufficiently small c > 0, ε > 0, for every graph G of order n with ⌈ (1− 1/r − ε)n2/2 ⌉ edges, one of the following conditions holds: (a) Kr+1 ( ⌊c lnn⌋...

We consider two extremal problems in hypergraphs. First, given k ≥ 3 and k-partite k-uniform hypergraphs, as a generalization of graph (k = 2) matchings, we determine the partite minimum codegree threshold for matchings with at most one vertex left in each part, thereby answering a problem asked by Rödl and Ruciński. We further improve the partite minimum codegree conditions to sum of all k par...

Tur\'an's Theorem says that an extremal $K_{r+1}$-free graph is $r$-partite. The Stability of Erd\H{o}s and Simonovits shows if a with $n$ vertices has close to the maximal $t_r(n)$ edges, then it being In this paper we determine exactly graphs at least $m$ edges are farthest from $r$-partite, for any $m\ge t_r(n) - \delta_r n^2$. This extends work by Erd\H{o}s, Gy\H{o}ri Simonovits, proves con...

Suppose D is an acyclic orientation of a graph G. An arc of D is said to be independent if its reversal results another acyclic orientation. Denote i(D) the number of independent arcs in D and N(G) = {i(D) : D is an acyclic orientation of G}. Also, let imin(G) be the minimum of N(G) and imax(G) the maximum. While it is known that imin(G) = |V (G)| − 1 for any connected graph G, the present pape...

graph is called a complete r-partite graph if its vertex set can be partitioned into r independent sets V1, . . . , Vr such that for any i = 1, 2, . . . , r every vertex in Vi is adjacent to all other vertices in Vj , j 6= i. We denote a complete r-partite graph with part sizes |Vi| = ni by Kn1,...,nr . For a graph G and a positive integer k we use kG to denote k vertex-disjoint copies of G. Gi...