Extremal graph theory is a branch of combinatorial mathematics that studies the extremal properties of graphs. Specifically, it focuses on questions related to the maximal or minimal number of edges in a graph that satisfies certain properties or conditions. The primary goal is often to determine the extremal (that is, maximum or minimum) values for specific parameters of graphs (like the number of edges, number of vertices, etc.) that meet certain constraints, such as containing a particular subgraph or avoiding certain configurations.
A **biclique-free graph** is a graph that does not contain any complete bipartite subgraph \( K_{m,n} \) as a subgraph. A complete bipartite graph \( K_{m,n} \) consists of two disjoint sets of vertices \( U \) and \( V \) where every vertex in \( U \) is connected to every vertex in \( V \), and there are no edges between vertices within the same set.
In the context of computer science and mathematics, a "common graph" can refer to different concepts depending on the specific area of discussion. However, it is not a universally defined or standard term.
Dependent random choice is a concept mainly used in probability theory and stochastic processes. It refers to a selection process where the choices made are not independent of one another; rather, the outcome of one choice influences the probabilities of subsequent choices. In a typical independent random choice scenario, the probability of each outcome remains constant regardless of what has happened before. However, in dependent random choice, the selection of one item or event alters the likelihood of selecting other items or events in the future.
The Forbidden Subgraph Problem is a concept from graph theory, related to understanding the structure of graphs by identifying certain subgraphs that are "forbidden" or not allowed within a graph. More formally, the problem can be described as follows: Given a graph \( G \) and a set of graphs \( H \), the Forbidden Subgraph Problem asks whether \( G \) contains any subgraph that is isomorphic to any of the graphs in the set \( H \).
Homomorphism density is a concept from combinatorics and graph theory that deals with the frequency of the occurrence of one graph within another graph. More formally, it relates to the density of homomorphisms from one graph to another.
The Turán graph, denoted as \( T(n, r) \), is a specific type of graph used in extremal graph theory, which studies the conditions under which graphs contain certain subgraphs. The Turán graph is designed to be the largest \( K_{r+1} \)-free graph (a graph that does not contain a complete subgraph of \( r+1 \) vertices) with \( n \) vertices.
The Turán number, denoted as \( T(n, r) \), is a concept in combinatorial mathematics, specifically in the field of graph theory. It represents the maximum number of edges in a graph with \( n \) vertices that does not contain any complete subgraph (or clique) with \( r \) vertices. In other words, it provides an upper limit on the edge count of a graph while avoiding certain cliques.
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