Basic Definitions and Concepts Clause Samples

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.