Basic Definitions and Concepts. We start this section with a review of various notions of quasirandomness for graphs, and will point out their relations. Our objects of interest in this chapter are simple graphs, without loops and multiple edges, and we refer the reader to [10] for basic graph theoretic definitions and concepts. Xxxxxxxx [37] was the first to study quasirandom graphs systematically. This research was continued by Xxxxx, Xxxxxx, and Xxxxxx [6] who investigated several properties of random-like graphs of density d = 1/2 and proved that they are all equivalent. For our purpose we will restrict ourselves to bipartite graphs here.
Basic Definitions and Concepts. We will start this section with some basic definitions. From now on we will only consider 3-uniform hypergraphs. Therefore, let us introduce some notation. We denote by [n] the set {1, . . . , n}. Suppose V is a set, then [V ]3 denotes all 3-element subsets of V . A 3-uniform hypergraph (also called 3-graph) H is a pair (V, E) where E(H) ⊆ [V ]3. The central objects in this thesis are tripartite 3-uniform hypergraphs with vertex partition V = V1 ∪ V2 ∪ V3, |V1| = m1, |V2| = m2, and |V3| = m3. As for graphs we also want to define quasirandomness for 3-uniform hypergraphs. Similar to Definition 2.1 where we considered labelled 4-cycles in a graph we define for tripartite 3-uniform hypergraphs a so-called octahedron O, where V (O) corresponds to (distinct) vertices v1, v1' ∈ V1, v2, v2' ∈ V2, and v3, v3' ∈ V3 2,2,2 that span an ‘ordered’ copy of K(3) , i.e., the complete tripartite 3-uniform hypergraph with vertex classes {v1, v1' }, {v2, v2' }, and {v3, v3' }. Our main objects of interests are so-called quasirandom 3-uniform hyper- graphs. You can find many different definitions of quasirandom 3-uniform hypergraphs in the literature. Here we will mention three definitions of a quasirandom 3-uniform hypergraph. These concepts can be viewed as gen- eralizations of the concepts for graphs discussed in Chapter 2. Moreover, as in Chapter 2 we will show that these definitions are all equivalent.
