Nam Hung Tran Nguyen

1. Why it is called "Ping-Pong Lemma"?       All the versions of "Ping-Pong Lemma" are constructed in the same manner: if you want to prove that a group is a free group, it is sufficient to prove that that group can "ping-pong" an element from a "region" to another "region" and ensure that it won't go back to the original "region". This is similar to the case when you "ping-pong" the ball when playing table tennis, where you need to move the ball from your side to the rival's side and prevent the ball from tracing back to your side.     This lemma is sometimes called the Schottky lemma (named after the German mathematician Friedrich Schottky) or Klein's criterion (named after the German mathematician Felix Klein).   2. Different versions of "Ping-Pong Lemma"    There have been various versions of "Ping-Pong Lemma" mentioned in different textbooks. As Johanna Mangahas wrote in "Off...

Note: This writing is/was written on March 10, 2025, which was the 61th birthday of the Czech mathematician Jiri Matousek - one of the two authors of "Invitation to Discrete Mathematics". Problem 27/77 was added on April 5, 2025. Problem 10/137 was added on April 6, 2025.  10/24: Let M be a set of real numbers, such that any nonempty subset of M has a smallest number and also a largest number. Prove that M is necessarily finite.  Solution:  Assume that M is infinite. Since M is a subset of itself, it has a smallest number A and a largest number B. Consider the sequence of sets M(0), M(1), ..., M(n), ... defined as follows:     i) M(0) = M;    ii) For each i, M(i + 1) is obtained by removing the smallest element (called a(i)) from M(i).     We have A = a(0) < a(1) < a(2) < ... Moreover, this sequence is bounded above by B, thus it has a limit L. Let X be the set of all numbers of the form a(n). Since the limit of this seque...