Category Discrete mathematics

Empty bins

Suppose that  balls,  where is a constant, are thrown sequentially to bins, all of which are initially empty. Let be the number of empty bins after we have thrown balls. Clearly, . Let’s see how we can understand this sequence of random variables. When we move from step to we throw a ball. It either falls in an […]

Graph sphericity

The sphericity  of a graph is the smallest dimension such that there exists a mapping such that if and only if . The following theorem is due to Frankl and Maehara  and is a nice application of random projections. Here, . Theorem [Frankl-Maehara 1988]: Let be a graph with minimum adjacency eigenvalue where and suppose . […]