On saturation of Berge hypergraphs

Forbidden Berge Hypergraphs

A simple matrix is a (0,1)-matrix with no repeated columns. For a (0,1)-matrix F , we say that a (0,1)-matrix A has F as a Berge hypergraph if there is a submatrix B of A and some row and column permutation of F , say G, with G 6 B. Letting ‖A‖ denote the number of columns in A, we define the extremal function Bh(m,F ) = max{‖A‖ : A m-rowed simple matrix and no Berge hypergraph F}. We determine...

Extremal Results for Berge Hypergraphs

Let G be a graph and H be a hypergraph both on the same vertex set. We say that a hypergraph H is a Berge-G if there is a bijection f : E(G) → E(H) such that for e ∈ E(G) we have e ⊂ f(e). This generalizes the established definitions of “Berge path” and “Berge cycle” to general graphs. For a fixed graph G we examine the maximum possible size (i.e. the sum of the cardinality of each edge) of a h...

Detachments of Hypergraphs I: The Berge-Johnson Problem

Monochromatic Hamiltonian t-tight Berge-cycles in hypergraphs

In any r-uniform hypergraph H for 2 ≤ t ≤ r we define an runiform t-tight Berge-cycle of length , denoted by C , as a sequence of distinct vertices v1, v2, . . . , v , such that for each set (vi , vi+1, . . . ,vi+t−1 ) of t consecutive vertices on the cycle, there is an edge Ei of H that contains these t vertices and the edges Ei are all distinct for i, 1 ≤ i ≤ , where + j ≡ j. For t = 2 we get...

Turán numbers for Berge-hypergraphs and related extremal problems

Let F be a graph. We say that a hypergraph H is a Berge-F if there is a bijection f : E(F )→ E(H) such that e ⊆ f(e) for every e ∈ E(F ). Note that Berge-F actually denotes a class of hypergraphs. The maximum number of edges in an n-vertex r-graph with no subhypergraph isomorphic to any Berge-F is denoted exr(n,Berge-F ). In this paper we establish new upper and lower bounds on exr(n,Berge-F ) ...