In mathematics, dependent random choice is a simple yet powerful probabilistic technique which shows how to find a large set of vertices in a dense graph such that every small subset of vertices has a lot of common neighbors. It is a useful tool to embed a graph into another graph with many edges, and thus has its application in extremal graph theory and Ramsey theory.
Statement of theorem
Let, and suppose Then every graph on vertices with at least edges contains a subset of vertices with such that for all with, has at least common neighbors.
Proof
The basic idea is to choose the set of vertices randomly. However, instead of choosing each vertex uniformly at random, the procedure chooses a list of vertices first and then chooses its common neighbors as the set of vertices. The hope is that in this way, the chosen set would be more likely to have more common neighbors. Formally, let be a list of vertices chosen uniformly at random from with replacement. Let be the common neighborhood of. The expected value of isFor every -element subset of, the event of containing occurs if and only if is contained in the common neighborhood of, which occurs with probability Call an bad if it has less than common neighbors. Then for each fixed -element subset of, it is contained in with probability less than. Therefore by linearity of expectation,To make sure that there are no bad subsets, we can get rid of one element in each bad subset. The number of remaining elements is at least, whose expected value is at least Consequently, there exists a such that there are at least elements in remaining after getting rid of all bad -element subsets. The set of the remaining elements satisfies the desired properties.
Directly using dependent random choice by setting appropriate parameters, we can show that if is a bipartite graph in which all vertices in have degree at most, then the extremal number where only depends on. Formally, if and is a sufficiently large constant such that Then by setting we know that and so the assumption of dependent random choice holds. Hence, for each graph with at least edges, there exists a vertex subset of size satisfying that every -subset of has at least common neighbors. This allows us to embed into in the following way. Embed into arbitrarily, and then embed the vertices in one by one. For each vertex in, it has at most neighbors in, which shows that their images in have at least common neighbors. This allows us to embed to one of the common neighbors while avoiding collisions. In fact, this can be generalized to degenerate graphs using the variation of dependent random choice described below.
Embedding a 1-subdivision">Homeomorphism_(graph_theory)#Subdivision_and_smoothing">1-subdivision of a complete graph
Dependent random choice can be applied directly to show that if is a graph on vertices and edges, then contains a 1-subdivision of a complete graph with vertices. This can be shown in a similar way to the above proof of the bound on Turán number of a bipartite graph. Indeed, if we set, we have and so the assumption of dependent random choice holds. Since a 1-subdivision of the complete graph on vertices is a bipartite graph with parts of size and where every vertex in the second part has degree two, we can run the embedding argument in the above proof of the bound on Turán number of a bipartite graph to get the desired result.
Variation
There is a stronger version of dependent random choice that finds two subsets of vertices in a dense graph so that every small subset of vertices in has a lot of common neighbors in. Formally, let be some positive integers with, and let be some real number. Suppose that the following constraints hold: Then every graph on vertices with at least edges contains two subsets of vertices so that any vertices in have at least common neighbors in.
Extremal number of a degenerate">Degeneracy_(graph_theory)">degenerate bipartite graph
Using this stronger statement, one can upper bound the extremal number of -degenerate bipartite graph: for each -degenerate bipartite graph with at most vertices, the extremal number ist at most
This statement can be also applied to obtain an upper bound of the Ramsey number of a degenerate bipartite graphs. If is a fixed integer, then for every biparpite -degenerate bipartite graph on vertices, the Ramsey number is of the order
